maths.free › Calculus › 2. Limits › The Limit of a Function

The Limit of a Function

Using correct notation, describe the limit of a function.

Intuitive Definition of a Limit

Let’s first take a closer look at how the function \(f(x)=({x}^{2}-4)\text{/}(x-2)\) behaves around \(x=2\) in . As the values of x approach 2 from either side of 2, the values of \(y=f(x)\) approach 4. Mathematically, we say that the limit of \(f(x)\) as x approaches 2 is 4. Symbolically, we express this limit as

\[\underset{x\to 2}{\text{lim}}f(x)=4.\]

From this very brief informal look at one limit, let’s start to develop an intuitive definition of the limit. We can think of the limit of a function at a number a as being the one real number L that the functional values approach as the x-values approach a, provided such a real number L exists. Stated more carefully, we have the following definition:

We can estimate limits by constructing tables of functional values and by looking at their graphs. This process is described in the following Problem-Solving Strategy.

We apply this Problem-Solving Strategy to compute a limit in .

Example

Try it.

Evaluate \(\underset{x\to 0}{\text{lim}}\frac{\text{sin}\ x}{x}\) using a table of functional values.

Solution

We have calculated the values of \(f(x)=(\text{sin}\ x)\text{/}x\) for the values of x listed in .

x\(\frac{\text{sin}\ x}{x}\)x\(\frac{\text{sin}\ x}{x}\)
−0.10.9983341664680.10.998334166468
−0.010.9999833334170.010.999983333417
−0.0010.9999998333330.0010.999999833333
−0.00010.9999999983330.00010.999999998333

Note: The values in this table were obtained using a calculator and using all the places given in the calculator output.

As we read down each \(\frac{(\text{sin}\ x)}{x}\) column, we see that the values in each column appear to be approaching one. Thus, it is fairly reasonable to conclude that \(\underset{x\to 0}{\text{lim}}\frac{\text{sin}\ x}{x}=1.\) A calculator or computer-generated graph of \(f(x)=\frac{(\text{sin}\ x)}{x}\) would be similar to that shown in , and it confirms our estimate.

x\(f(x)=c\)x\(f(x)=c\)
\(a-0.1\)c\(a+0.1\)c
\(a-0.01\)c\(a+0.01\)c
\(a-0.001\)c\(a+0.001\)c
\(a-0.0001\)c\(a+0.0001\)c

Condensed — the full section is in OpenStax Calculus Volume 1.

The Existence of a Limit

As we consider the limit in the next example, keep in mind that for the limit of a function to exist at a point, the functional values must approach a single real-number value at that point. If the functional values do not approach a single value, then the limit does not exist.

Example

Try it.

Evaluate \(\underset{x\to 0}{\text{lim}}\text{sin}\ (1\text{/}\text{x})\) using a table of values.

Solution

lists values for the function \(\text{sin}(1\text{/}x)\) for the given values of x.

x\(\text{sin}(\frac{1}{x})\)x\(\text{sin}(\frac{1}{x})\)
−0.10.5440211108890.1−0.544021110889
−0.010.506365641110.01−0.50636564111
−0.001−0.82687954053120.0010.826879540532
−0.00010.3056143888880.0001−0.305614388888
−0.00001−0.0357487979870.000010.035748797987
−0.0000010.3499935041870.000001−0.349993504187

After examining the table of functional values, we can see that the y-values do not seem to approach any one single value. It appears the limit does not exist. Before drawing this conclusion, let’s take a more systematic approach. Take the following sequence of x-values approaching 0:

\[\frac{2}{\pi },\frac{2}{3\pi },\frac{2}{5\pi },\frac{2}{7\pi },\frac{2}{9\pi },\frac{2}{11\pi }\text{,\ldots .}\]

The corresponding y-values are

\[1,-1,1,-1,1,-1\text{,\ldots .}\]

At this point we can indeed conclude that \(\underset{x\to 0}{\text{lim}}\text{sin}\ (1\text{/}\text{x})\) does not exist. (Mathematicians frequently abbreviate “does not exist” as DNE. Thus, we would write \(\underset{x\to 0}{\text{lim}}\text{sin}\ (1\text{/}\text{x})\) DNE.) The graph of \(f(x)=\text{sin}\ (1\text{/}x)\) is shown in and it gives a clearer picture of the behavior of \(\text{sin}(1\text{/}x)\) as x approaches 0. You can see that \(\text{sin}(1\text{/}\text{x})\) oscillates ever more wildly between −1 and 1 as x approaches 0.

One-Sided Limits

Sometimes indicating that the limit of a function fails to exist at a point does not provide us with enough information about the behavior of the function at that particular point. To see this, we now revisit the function \(g(x)=|x-2|\text{/}(x-2)\) introduced at the beginning of the section (see (b)). As we pick values of x close to 2, \(g(x)\) does not approach a single value, so the limit as x approaches 2 does not exist—that is, \(\underset{x\to 2}{\text{lim}}g(x)\) DNE. However, this statement alone does not give us a complete picture of the behavior of the function around the x-value 2. To provide a more accurate description, we introduce the idea of a one-sided limit. For all values to the left of 2 (or the negative side of 2), \(g(x)=-1.\) Thus, as x approaches 2 from the left, \(g(x)\) approaches −1. Mathematically, we say that the limit as x approaches 2 from the left is −1. Symbolically, we express this idea as

\[\underset{x\to {2}^{-}}{\text{lim}}g(x)=-1.\]

Similarly, as x approaches 2 from the right (or from the positive side), \(g(x)\) approaches 1. Symbolically, we express this idea as

\[\underset{x\to {2}^{+}}{\text{lim}}g(x)=1.\]

We can now present an informal definition of one-sided limits.

Condensed — the full section is in OpenStax Calculus Volume 1.

Infinite Limits

Evaluating the limit of a function at a point or evaluating the limit of a function from the right and left at a point helps us to characterize the behavior of a function around a given value. As we shall see, we can also describe the behavior of functions that do not have finite limits.

We now turn our attention to \(h(x)=1\text{/}{(x-2)}^{2},\) the third and final function introduced at the beginning of this section (see (c)). From its graph we see that as the values of x approach 2, the values of \(h(x)=1\text{/}{(x-2)}^{2}\) become larger and larger and, in fact, become infinite. Mathematically, we say that the limit of \(h(x)\) as x approaches 2 is positive infinity. Symbolically, we express this idea as

\[\underset{x\to 2}{\text{lim}}h(x)=\text{+}\infty .\]

More generally, we define infinite limits as follows:

It is important to understand that when we write statements such as \(\underset{x\to a}{\text{lim}}f(x)=\text{+}\infty\) or \(\underset{x\to a}{\text{lim}}f(x)=\text{-}\infty\) we are describing the behavior of the function, as we have just defined it. We are not asserting that a limit exists. For the limit of a function \(f(x)\) to exist at a, it must approach a real number L as x approaches a. That said, if, for example, \(\underset{x\to a}{\text{lim}}f(x)=\text{+}\infty ,\) we always write \(\underset{x\to a}{\text{lim}}f(x)=\text{+}\infty\) rather than \(\underset{x\to a}{\text{lim}}f(x)\) DNE.

Example

Try it.

Evaluate each of the following limits, if possible. Use a table of functional values and graph \(f(x)=1\text{/}x\) to confirm your conclusion.

  1. \(\underset{x\to {0}^{-}}{\text{lim}}\frac{1}{x}\)
  2. \(\underset{x\to {0}^{+}}{\text{lim}}\frac{1}{x}\)
  3. \(\underset{x\to 0}{\text{lim}}\frac{1}{x}\)
Solution

Begin by constructing a table of functional values.

x\(\frac{1}{x}\)x\(\frac{1}{x}\)
−0.1−100.110
−0.01−1000.01100
−0.001−10000.0011000
−0.0001−10,0000.000110,000
−0.00001−100,0000.00001100,000
−0.000001−1,000,0000.0000011,000,000
  1. The values of \(1\text{/}x\) decrease without bound as x approaches 0 from the left. We conclude that
    \[\underset{x\to {0}^{-}}{\text{lim}}\frac{1}{x}=\text{-}\infty .\]
  2. The values of \(1\text{/}x\) increase without bound as x approaches 0 from the right. We conclude that
    \[\underset{x\to {0}^{+}}{\text{lim}}\frac{1}{x}=\text{+}\infty .\]
  3. Since \(\underset{x\to {0}^{-}}{\text{lim}}\frac{1}{x}=\text{-}\infty\) and \(\underset{x\to {0}^{+}}{\text{lim}}\frac{1}{x}=\text{+}\infty\) have different values, we conclude that
    \[\underset{x\to 0}{\text{lim}}\frac{1}{x}\ \text{DNE.}\]

The graph of \(f(x)=1\text{/}x\) in confirms these conclusions.

Condensed — the full section is in OpenStax Calculus Volume 1.

Key Concepts

  • A table of values or graph may be used to estimate a limit.
  • If the limit of a function at a point does not exist, it is still possible that the limits from the left and right at that point may exist.
  • If the limits of a function from the left and right exist and are equal, then the limit of the function is that common value.
  • We may use limits to describe infinite behavior of a function at a point.

Key Equations

Intuitive Definition of the Limit\(\underset{x\to a}{\text{lim}}f(x)=L\)
Two Important Limits\(\underset{x\to a}{\text{lim}}x=a\ \underset{x\to a}{\text{lim}}c=c\)
One-Sided Limits\(\underset{x\to {a}^{-}}{\text{lim}}f(x)=L\ \underset{x\to {a}^{+}}{\text{lim}}f(x)=L\)
Infinite Limits from the Left\(\underset{x\to {a}^{-}}{\text{lim}}f(x)=\text{+}\infty \ \underset{x\to {a}^{-}}{\text{lim}}f(x)=\text{-}\infty\)
Infinite Limits from the Right\(\underset{x\to {a}^{+}}{\text{lim}}f(x)=\text{+}\infty \ \underset{x\to {a}^{+}}{\text{lim}}f(x)=\text{-}\infty\)
Two-Sided Infinite Limits\(\underset{x\to a}{\text{lim}}f(x)=\text{+}\infty :\underset{x\to {a}^{-}}{\text{lim}}f(x)=\text{+}\infty\) and \(\underset{x\to {a}^{+}}{\text{lim}}f(x)=\text{+}\infty\)
\(\underset{x\to a}{\text{lim}}f(x)=\text{-}\infty :\underset{x\to {a}^{-}}{\text{lim}}f(x)=\text{-}\infty\) and \(\underset{x\to {a}^{+}}{\text{lim}}f(x)=\text{-}\infty\)

The Limit of a Function

For the following exercises, consider the function \(f(x)=\frac{{x}^{2}-1}{|x-1|}.\)

For the following exercises, consider the function \(f(x)={(1+x)}^{1\text{/}x}.\)

In the following exercises, use the given values to set up a table to evaluate the limits. Round your solutions to eight decimal places.

[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits.

[T] In the following exercises, set up a table of values and round to eight significant digits. Based on the table of values, make a guess about what the limit is. Then, use a calculator to graph the function and determine the limit. Was the conjecture correct? If not, why does the method of tables fail?

In the following exercises, consider the graph of the function \(y=f(x)\) shown here. Which of the statements about \(y=f(x)\) are true and which are false? Explain why a statement is false.

In the following exercises, use the following graph of the function \(y=f(x)\) to find the values, if possible. Estimate when necessary.

Condensed — the full section is in OpenStax Calculus Volume 1.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Evaluate \(\underset{x\to 0}{\text{lim}}\frac{\text{sin}\ x}{x}\) using a table of functional values.

    Jawaby görkez

    We have calculated the values of \(f(x)=(\text{sin}\ x)\text{/}x\) for the values of x listed in .

    x\(\frac{\text{sin}\ x}{x}\)x\(\frac{\text{sin}\ x}{x}\)
    −0.10.9983341664680.10.998334166468
    −0.010.9999833334170.010.999983333417
    −0.0010.9999998333330.0010.999999833333
    −0.00010.9999999983330.00010.999999998333

    Note: The values in this table were obtained using a calculator and using all the places given in the calculator output.

    As we read down each \(\frac{(\text{sin}\ x)}{x}\) column, we see that the values in each column appear to be approaching one. Thus, it is fairly reasonable to conclude that \(\underset{x\to 0}{\text{lim}}\frac{\text{sin}\ x}{x}=1.\) A calculator or computer-generated graph of \(f(x)=\frac{(\text{sin}\ x)}{x}\) would be similar to that shown in , and it confirms our estimate.

  2. Evaluate \(\underset{x\to 4}{\text{lim}}\frac{\sqrt{x}-2}{x-4}\) using a table of functional values.

    Jawaby görkez

    As before, we use a table—in this case, —to list the values of the function for the given values of x.

    x\(\frac{\sqrt{x}-2}{x-4}\)x\(\frac{\sqrt{x}-2}{x-4}\)
    3.90.2515823418694.10.248456731317
    3.990.250156445624.010.24984394501
    3.9990.2500156274.0010.249984377
    3.99990.2500015634.00010.249998438
    3.999990.250000164.000010.24999984

    After inspecting this table, we see that the functional values less than 4 appear to be decreasing toward 0.25 whereas the functional values greater than 4 appear to be increasing toward 0.25. We conclude that \(\underset{x\to 4}{\text{lim}}\frac{\sqrt{x}-2}{x-4}=0.25.\) We confirm this estimate using the graph of \(f(x)=\frac{\sqrt{x}-2}{x-4}\) shown in .

  3. Estimate \(\underset{x\to 1}{\text{lim}}\frac{\frac{1}{x}-1}{x-1}\) using a table of functional values. Use a graph to confirm your estimate.

    Jawaby görkez

    \(\underset{x\to 1}{\text{lim}}\frac{\frac{1}{x}-1}{x-1}=-1\)

  4. For \(g(x)\) shown in , evaluate \(\underset{x\to -1}{\text{lim}}g(x).\)

    Jawaby görkez

    Despite the fact that \(g(-1)=4,\) as the x-values approach −1 from either side, the \(g(x)\) values approach 3. Therefore, \(\underset{x\to -1}{\text{lim}}g(x)=3.\) Note that we can determine this limit without even knowing the algebraic expression of the function.

  5. Use the graph of \(h(x)\) in to evaluate \(\underset{x\to 2}{\text{lim}}h(x),\) if possible.

    Jawaby görkez

    \(\underset{x\to 2}{\text{lim}}h(x)=-1.\)

  6. Evaluate \(\underset{x\to 0}{\text{lim}}\text{sin}\ (1\text{/}\text{x})\) using a table of values.

    Jawaby görkez

    lists values for the function \(\text{sin}(1\text{/}x)\) for the given values of x.

    x\(\text{sin}(\frac{1}{x})\)x\(\text{sin}(\frac{1}{x})\)
    −0.10.5440211108890.1−0.544021110889
    −0.010.506365641110.01−0.50636564111
    −0.001−0.82687954053120.0010.826879540532
    −0.00010.3056143888880.0001−0.305614388888
    −0.00001−0.0357487979870.000010.035748797987
    −0.0000010.3499935041870.000001−0.349993504187

    After examining the table of functional values, we can see that the y-values do not seem to approach any one single value. It appears the limit does not exist. Before drawing this conclusion, let’s take a more systematic approach. Take the following sequence of x-values approaching 0:

    \[\frac{2}{\pi },\frac{2}{3\pi },\frac{2}{5\pi },\frac{2}{7\pi },\frac{2}{9\pi },\frac{2}{11\pi }\text{,\ldots .}\]

    The corresponding y-values are

    \[1,-1,1,-1,1,-1\text{,\ldots .}\]

    At this point we can indeed conclude that \(\underset{x\to 0}{\text{lim}}\text{sin}\ (1\text{/}\text{x})\) does not exist. (Mathematicians frequently abbreviate “does not exist” as DNE. Thus, we would write \(\underset{x\to 0}{\text{lim}}\text{sin}\ (1\text{/}\text{x})\) DNE.) The graph of \(f(x)=\text{sin}\ (1\text{/}x)\) is shown in and it gives a clearer picture of the behavior of \(\text{sin}(1\text{/}x)\) as x approaches 0. You can see that \(\text{sin}(1\text{/}\text{x})\) oscillates ever more wildly between −1 and 1 as x approaches 0.

  7. Use a table of functional values to evaluate \(\underset{x\to 2}{\text{lim}}\frac{|{x}^{2}-4|}{x-2},\) if possible.

    Jawaby görkez

    \(\underset{x\to 2}{\text{lim}}\frac{|{x}^{2}-4|}{x-2}\) does not exist.

  8. For the function \(f(x)=\{\begin{array}{ll}x+1 & \text{if}\ x<2 \\ {x}^{2}-4 & \text{if}\ x\ge 2\end{array},\) evaluate each of the following limits.

    1. \(\underset{x\to {2}^{-}}{\text{lim}}f(x)\)
    2. \(\underset{x\to {2}^{+}}{\text{lim}}f(x)\)
    Jawaby görkez

    We can use tables of functional values again . Observe that for values of x less than 2, we use \(f(x)=x+1\) and for values of x greater than 2, we use \(f(x)={x}^{2}-4.\)

    x\(f(x)=x+1\)x\(f(x)={x}^{2}-4\)
    1.92.92.10.41
    1.992.992.010.0401
    1.9992.9992.0010.004001
    1.99992.99992.00010.00040001
    1.999992.999992.000010.0000400001

    Based on this table, we can conclude that a. \(\underset{x\to {2}^{-}}{\text{lim}}f(x)=3\) and b. \(\underset{x\to {2}^{+}}{\text{lim}}f(x)=0.\) Therefore, the (two-sided) limit of \(f(x)\) does not exist at \(x=2.\) shows a graph of \(f(x)\) and reinforces our conclusion about these limits.

  9. Use a table of functional values to estimate the following limits, if possible.

    1. \(\underset{x\to {2}^{-}}{\text{lim}}\frac{|{x}^{2}-4|}{x-2}\)
    2. \(\underset{x\to {2}^{+}}{\text{lim}}\frac{|{x}^{2}-4|}{x-2}\)
    Jawaby görkez

    a. \(\underset{x\to {2}^{-}}{\text{lim}}\frac{|{x}^{2}-4|}{x-2}=-4;\) b. \(\underset{x\to {2}^{+}}{\text{lim}}\frac{|{x}^{2}-4|}{x-2}=4\)

  10. Evaluate each of the following limits, if possible. Use a table of functional values and graph \(f(x)=1\text{/}x\) to confirm your conclusion.

    1. \(\underset{x\to {0}^{-}}{\text{lim}}\frac{1}{x}\)
    2. \(\underset{x\to {0}^{+}}{\text{lim}}\frac{1}{x}\)
    3. \(\underset{x\to 0}{\text{lim}}\frac{1}{x}\)
    Jawaby görkez

    Begin by constructing a table of functional values.

    x\(\frac{1}{x}\)x\(\frac{1}{x}\)
    −0.1−100.110
    −0.01−1000.01100
    −0.001−10000.0011000
    −0.0001−10,0000.000110,000
    −0.00001−100,0000.00001100,000
    −0.000001−1,000,0000.0000011,000,000
    1. The values of \(1\text{/}x\) decrease without bound as x approaches 0 from the left. We conclude that
      \[\underset{x\to {0}^{-}}{\text{lim}}\frac{1}{x}=\text{-}\infty .\]
    2. The values of \(1\text{/}x\) increase without bound as x approaches 0 from the right. We conclude that
      \[\underset{x\to {0}^{+}}{\text{lim}}\frac{1}{x}=\text{+}\infty .\]
    3. Since \(\underset{x\to {0}^{-}}{\text{lim}}\frac{1}{x}=\text{-}\infty\) and \(\underset{x\to {0}^{+}}{\text{lim}}\frac{1}{x}=\text{+}\infty\) have different values, we conclude that
      \[\underset{x\to 0}{\text{lim}}\frac{1}{x}\ \text{DNE.}\]

    The graph of \(f(x)=1\text{/}x\) in confirms these conclusions.

  11. Evaluate each of the following limits, if possible. Use a table of functional values and graph \(f(x)=1\text{/}{x}^{2}\) to confirm your conclusion.

    1. \(\underset{x\to {0}^{-}}{\text{lim}}\frac{1}{{x}^{2}}\)
    2. \(\underset{x\to {0}^{+}}{\text{lim}}\frac{1}{{x}^{2}}\)
    3. \(\underset{x\to 0}{\text{lim}}\frac{1}{{x}^{2}}\)
    Jawaby görkez

    a. \(\underset{x\to {0}^{-}}{\text{lim}}\frac{1}{{x}^{2}}=\text{+}\infty ;\) b. \(\underset{x\to {0}^{+}}{\text{lim}}\frac{1}{{x}^{2}}=\text{+}\infty ;\) c. \(\underset{x\to 0}{\text{lim}}\frac{1}{{x}^{2}}=\text{+}\infty\)

  12. Evaluate each of the following limits using . Identify any vertical asymptotes of the function \(f(x)=1\text{/}{(x+3)}^{4}.\)

    1. \(\underset{x\to {-3}^{-}}{\text{lim}}\frac{1}{{(x+3)}^{4}}\)
    2. \(\underset{x\to {-3}^{+}}{\text{lim}}\frac{1}{{(x+3)}^{4}}\)
    3. \(\underset{x\to -3}{\text{lim}}\frac{1}{{(x+3)}^{4}}\)
    Jawaby görkez

    We can use directly.

    1. \(\underset{x\to {-3}^{-}}{\text{lim}}\frac{1}{{(x+3)}^{4}}=\text{+}\infty\)
    2. \(\underset{x\to {-3}^{+}}{\text{lim}}\frac{1}{{(x+3)}^{4}}=\text{+}\infty\)
    3. \(\underset{x\to -3}{\text{lim}}\frac{1}{{(x+3)}^{4}}=\text{+}\infty\)

    The function \(f(x)=1\text{/}{(x+3)}^{4}\) has a vertical asymptote of \(x=-3.\)

  13. Evaluate each of the following limits. Identify any vertical asymptotes of the function \(f(x)=\frac{1}{{(x-2)}^{3}}.\)

    1. \(\underset{x\to {2}^{-}}{\text{lim}}\frac{1}{{(x-2)}^{3}}\)
    2. \(\underset{x\to {2}^{+}}{\text{lim}}\frac{1}{{(x-2)}^{3}}\)
    3. \(\underset{x\to 2}{\text{lim}}\frac{1}{{(x-2)}^{3}}\)
    Jawaby görkez

    a. \(\underset{x\to {2}^{-}}{\text{lim}}\frac{1}{{(x-2)}^{3}}=\text{-}\infty ;\) b. \(\underset{x\to {2}^{+}}{\text{lim}}\frac{1}{{(x-2)}^{3}}=\text{+}\infty ;\) c. \(\underset{x\to 2}{\text{lim}}\frac{1}{{(x-2)}^{3}}\) DNE. The line \(x=2\) is the vertical asymptote of \(f(x)=1\text{/}{(x-2)}^{3}.\)

  14. Use the graph of \(f(x)\) in to determine each of the following values:

    1. \(\underset{x\to {-4}^{-}}{\text{lim}}f(x);\underset{x\to {-4}^{+}}{\text{lim}}f(x);\underset{x\to -4}{\text{lim}}f(x);f(-4)\)
    2. \(\underset{x\to {-2}^{-}}{\text{lim}}f(x);\underset{x\to {-2}^{+}}{\text{lim}}f(x);\underset{x\to -2}{\text{lim}}f(x);f(-2)\)
    3. \(\underset{x\to {1}^{-}}{\text{lim}}f(x);\underset{x\to {1}^{+}}{\text{lim}}f(x);\underset{x\to 1}{\text{lim}}f(x);f(1)\)
    4. \(\underset{x\to {3}^{-}}{\text{lim}}f(x);\underset{x\to {3}^{+}}{\text{lim}}f(x);\underset{x\to 3}{\text{lim}}f(x);f(3)\)
    Jawaby görkez

    Using and the graph for reference, we arrive at the following values:

    1. \(\underset{x\to {-4}^{-}}{\text{lim}}f(x)=0;\underset{x\to {-4}^{+}}{\text{lim}}f(x)=0;\underset{x\to -4}{\text{lim}}f(x)=0;f(-4)=0\)
    2. \(\underset{x\to {-2}^{-}}{\text{lim}}f(x)=3.;\underset{x\to {-2}^{+}}{\text{lim}}f(x)=3;\underset{x\to -2}{\text{lim}}f(x)=3;f(-2)\) is undefined
    3. \(\underset{x\to {1}^{-}}{\text{lim}}f(x)=6;\underset{x\to {1}^{+}}{\text{lim}}f(x)=3;\underset{x\to 1}{\text{lim}}f(x)\) DNE; \(f(1)=6\)
    4. \(\underset{x\to {3}^{-}}{\text{lim}}f(x)=\text{-}\infty ;\underset{x\to {3}^{+}}{\text{lim}}f(x)=\text{-}\infty ;\underset{x\to 3}{\text{lim}}f(x)=\text{-}\infty ;f(3)\) is undefined

  15. Evaluate \(\underset{x\to 1}{\text{lim}}f(x)\) for \(f(x)\) shown here:

    Jawaby görkez

    Does not exist.

  16. In the chapter opener we mentioned briefly how Albert Einstein showed that a limit exists to how fast any object can travel. Given Einstein’s equation for the mass of a moving object, what is the value of this bound?

    Jawaby görkez

    Our starting point is Einstein’s equation for the mass of a moving object,

    \[m=\frac{{m}_{0}}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}},\]

    where \({m}_{0}\) is the object’s mass at rest, v is its speed, and c is the speed of light. To see how the mass changes at high speeds, we can graph the ratio of masses \(m\text{/}{m}_{0}\) as a function of the ratio of speeds, \(v\text{/}c\) ().

    We can see that as the ratio of speeds approaches 1—that is, as the speed of the object approaches the speed of light—the ratio of masses increases without bound. In other words, the function has a vertical asymptote at \(v\text{/}c=1.\) We can try a few values of this ratio to test this idea.

    \(\frac{v}{c}\)\(\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}\)\(\frac{m}{{m}_{0}}\)
    0.990.14117.089
    0.9990.044722.37
    0.99990.014170.71

    Thus, according to , if an object with mass 100 kg is traveling at 0.9999c, its mass becomes 7071 kg. Since no object can have an infinite mass, we conclude that no object can travel at or more than the speed of light.

  17. [T] Complete the following table for the function. Round your solutions to four decimal places.

    x\(f(x)\)x\(f(x)\)
    0.9a.1.1e.
    0.99b.1.01f.
    0.999c.1.001g.
    0.9999d.1.0001h.

  18. What do your results in the preceding exercise indicate about the two-sided limit \(\underset{x\to 1}{\text{lim}}f(x)?\) Explain your response.

    Jawaby görkez

    \(\underset{x\to 1}{\text{lim}}f(x)\) does not exist because \(\underset{x\to {1}^{-}}{\text{lim}}f(x)=-2\ne \underset{x\to {1}^{+}}{\text{lim}}f(x)=2.\)

  19. [T] Make a table showing the values of f for \(x=-0.01,-0.001,-0.0001,-0.00001\) and for \(x=0.01,0.001,0.0001,0.00001.\) Round your solutions to five decimal places.

    x\(f(x)\)x\(f(x)\)
    −0.01a.0.01e.
    −0.001b.0.001f.
    −0.0001c.0.0001g.
    −0.00001d.0.00001h.

  20. What does the table of values in the preceding exercise indicate about the function \(f(x)={(1+x)}^{1\text{/}x}?\)

    Jawaby görkez

    \(\underset{x\to 0}{\text{lim}}{(1+x)}^{1\text{/}x}=2.7183\)

  21. To which mathematical constant does the limit in the preceding exercise appear to be getting closer?

  22. [T] \(\underset{x\to 0}{\text{lim}}\frac{\text{sin}\ 2x}{x};\ \pm 0.1,\pm 0.01,\pm 0.001,\pm .0001\)

    x\(\frac{\text{sin}\ 2x}{x}\)x\(\frac{\text{sin}\ 2x}{x}\)
    −0.1a.0.1e.
    −0.01b.0.01f.
    −0.001c.0.001g.
    −0.0001d.0.0001h.
    Jawaby görkez

    a. 1.98669331; b. 1.99986667; c. 1.99999867; d. 1.99999999; e. 1.98669331; f. 1.99986667; g. 1.99999867; h. 1.99999999; \(\underset{x\to 0}{\text{lim}}\frac{\text{sin}\ 2x}{x}=2\)

  23. [T] \(\underset{x\to 0}{\text{lim}}\frac{\text{sin}\ 3x}{x}\) ±0.1, ±0.01, ±0.001, ±0.0001

    X\(\frac{\text{sin}\ 3x}{x}\)x\(\frac{\text{sin}\ 3x}{x}\)
    −0.1a.0.1e.
    −0.01b.0.01f.
    −0.001c.0.001g.
    −0.0001d.0.0001h.

  24. Use the preceding two exercises to conjecture (guess) the value of the following limit: \(\underset{x\to 0}{\text{lim}}\frac{\text{sin}\ ax}{x}\) for a, a positive real value.

    Jawaby görkez

    \(\underset{x\to 0}{\text{lim}}\frac{\text{sin}\ ax}{x}=a\)

  25. \(\underset{x\to 2}{\text{lim}}\frac{{x}^{2}-4}{{x}^{2}+x-6}\)

    x\(\frac{{x}^{2}-4}{{x}^{2}+x-6}\)x\(\frac{{x}^{2}-4}{{x}^{2}+x-6}\)
    1.9a.2.1e.
    1.99b.2.01f.
    1.999c.2.001g.
    1.9999d.2.0001h.

  26. \(\underset{x\to 1}{\text{lim}}(1-2x)\)

    x\(1-2x\)x\(1-2x\)
    0.9a.1.1e.
    0.99b.1.01f.
    0.999c.1.001g.
    0.9999d.1.0001h.
    Jawaby görkez

    a. −0.80000000; b. −0.98000000; c. −0.99800000; d. −0.99980000; e. −1.2000000; f. −1.0200000; g. −1.0020000; h. −1.0002000; \(\underset{x\to 1}{\text{lim}}(1-2x)=-1\)

  27. \(\underset{x\to 0}{\text{lim}}\frac{5}{1-{e}^{1\text{/}x}}\)

    x\(\frac{5}{1-{e}^{1\text{/}x}}\)x\(\frac{5}{1-{e}^{1\text{/}x}}\)
    −0.1a.0.1e.
    −0.01b.0.01f.
    −0.001c.0.001g.
    −0.0001d.0.0001h.

  28. \(\underset{z\to 0}{\text{lim}}\frac{z-1}{{z}^{2}(z+3)}\)

    z\(\frac{z-1}{{z}^{2}(z+3)}\)z\(\frac{z-1}{{z}^{2}(z+3)}\)
    −0.1a.0.1e.
    −0.01b.0.01f.
    −0.001c.0.001g.
    −0.0001d.0.0001h.
    Jawaby görkez

    a. −37.931934; b. −3377.9264; c. −333,777.93; d. −33,337,778; e. −29.032258; f. −3289.0365; g. −332,889.04; h. −33,328,889 \(\underset{x\to 0}{\text{lim}}\frac{z-1}{{z}^{2}(z+3)}=\text{-}\infty\)

  29. \(\underset{t\to {0}^{+}}{\text{lim}}\frac{\text{cos}\ t}{t}\)

    t\(\frac{\text{cos}\ t}{t}\)
    0.1a.
    0.01b.
    0.001c.
    0.0001d.

  30. \(\underset{x\to 2}{\text{lim}}\frac{1-\frac{2}{x}}{{x}^{2}-4}\)

    x\(\frac{1-\frac{2}{x}}{{x}^{2}-4}\)x\(\frac{1-\frac{2}{x}}{{x}^{2}-4}\)
    1.9a.2.1e.
    1.99b.2.01f.
    1.999c.2.001g.
    1.9999d.2.0001h.
    Jawaby görkez

    a. 0.13495277; b. 0.12594300; c. 0.12509381; d. 0.12500938; e. 0.11614402; f. 0.12406794; g. 0.12490631; h. 0.12499063; \(∴\underset{x\to 2}{\text{lim}}\frac{1-\frac{2}{x}}{{x}^{2}-4}=0.1250=\frac{1}{8}\)

  31. \(\underset{\theta \to 0}{\text{lim}}\text{sin}\ (\frac{\pi }{\theta })\)

    θ\(\text{sin}\ (\frac{\pi }{\theta })\)θ\(\text{sin}\ (\frac{\pi }{\theta })\)
    −0.1a.0.1e.
    −0.01b.0.01f.
    −0.001c.0.001g.
    −0.0001d.0.0001h.

  32. \(\underset{\alpha \to {0}^{+}}{\text{lim}}\frac{1}{\alpha }\ \text{cos}\ (\frac{\pi }{\alpha })\)

    \(\alpha\)\(\frac{1}{\alpha }\ \text{cos}\ (\frac{\pi }{\alpha })\)
    0.1a.
    0.01b.
    0.001c.
    0.0001d.
    Jawaby görkez

    a. 10.00000; b. 100.00000; c. 1000.0000; d. 10,000.000; Guess: \(\underset{\alpha \to {0}^{+}}{\text{lim}}\frac{1}{\alpha }\ \text{cos}\ (\frac{\pi }{\alpha })=\infty ,\) actual: DNE

  33. \(\underset{x\to 10}{\text{lim}}f(x)=0\)

  34. \(\underset{x\to {-2}^{+}}{\text{lim}}f(x)=3\)

    Jawaby görkez

    False; \(\underset{x\to {-2}^{+}}{\text{lim}}f(x)=\text{+}\infty\)

  35. \(\underset{x\to -8}{\text{lim}}f(x)=f(-8)\)

  36. \(\underset{x\to 6}{\text{lim}}f(x)=5\)

    Jawaby görkez

    False; \(\underset{x\to 6}{\text{lim}}f(x)\) DNE since \(\underset{x\to {6}^{-}}{\text{lim}}f(x)=2\) and \(\underset{x\to {6}^{+}}{\text{lim}}f(x)=5.\)

  37. \(\underset{x\to {1}^{-}}{\text{lim}}f(x)\)

  38. \(\underset{x\to {1}^{+}}{\text{lim}}f(x)\)

    Jawaby görkez

    2

  39. \(\underset{x\to 1}{\text{lim}}f(x)\)

  40. \(\underset{x\to 2}{\text{lim}}f(x)\)

    Jawaby görkez

    1

Symbols used here

\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
\infty
infinity
Not a number: "grows without bound" in limits and intervals.
\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\neq
not equal
The two sides are different.
\sup,\ \inf
supremum, infimum
Least upper bound, greatest lower bound.
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
\sin,\ \cos,\ \tan
sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\lim_{x \to a} f(x)
limit
The value f(x) approaches as x approaches a.
f'(x),\ \frac{dy}{dx}
derivative
Instantaneous rate of change; slope of the graph.
\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
y',\ y''
first and second derivative of y
Prime notation for derivatives with respect to x (or t).
C,\ C_1,\ C_2
arbitrary constants
Constants of integration fixed by initial conditions.

How to: The Limit of a Function

  1. Using correct notation, describe the limit of a function.
  2. Use a table of values to estimate the limit of a function or to identify when the limit does not exist.
  3. Use a graph to estimate the limit of a function or to identify when the limit does not exist.
  4. Define one-sided limits and provide examples.
  5. Explain the relationship between one-sided and two-sided limits.
  6. Using correct notation, describe an infinite limit.
  7. Define a vertical asymptote.
  8. To evaluate

Questions people ask

What is a derivative in one sentence?

The slope of the graph at a point — the rate at which the output is changing there. Speed is the derivative of position.

What is an integral in one sentence?

The accumulated total of a rate — the area under the curve. Distance travelled is the integral of speed.

Why are derivatives and integrals opposites?

That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.

When do I use substitution and when integration by parts?

Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function — a polynomial times an exponential, log or trig function.

Özüňi synla

Parts of this page are adapted from OpenStax Calculus Volume 1 (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

_Ýaşa Calculus

LimitsDerivativesIntegralsDefinite integralsTaylor seriesSeries and sumsMaxima and minimaThe chain ruleImplicit differentiationRelated rates and optimisationIntegration techniques: substitution, parts, partial fractionsApplications of integration: area, volume, arc lengthInfinite series and convergence tests