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Finding Limits: Numerical and Graphical Approaches

Understand limit notation.

Understanding Limit Notation

We have seen how a sequence can have a limit, a value that the sequence of terms moves toward as the number of terms increases. For example, the terms of the sequence

\[1,\frac{1}{2},\frac{1}{4},\frac{1}{8}...\]

gets closer and closer to 0. A sequence is one type of function, but functions that are not sequences can also have limits. We can describe the behavior of the function as the input values get close to a specific value. If the limit of a function \(f(x)=L\text{,}\) then as the input \(x\) gets closer and closer to \(a,\) the output y-coordinate gets closer and closer to \(L.\) We say that the output “approaches” \(L.\)

provides a visual representation of the mathematical concept of limit. As the input value \(x\) approaches \(a,\) the output value \(f(x)\) approaches \(L.\)

We write the equation of a limit as

\[\underset{x\to a}{\lim }f(x)=L.\]

This notation indicates that as \(x\) approaches \(a\) both from the left of \(x=a\) and the right of \(x=a,\) the output value approaches \(L.\)

Consider the function

\[f(x)=\frac{{x}^{2}-6x-7}{x-7}.\]

We can factor the function as shown.

\[\begin{array}{ll}f(x)=\frac{(x-7)(x+1)}{x-7}\ & \text{Cancel like factors in numerator and denominator.} \\ f(x)=x+1,x\ne 7 & \text{Simplify.}\end{array}\]\[\underset{x\to \ 7}{\lim }f(x)=8\]\[f(7)\text{ does not exist.}\]\[f(x)=x+1,\ x\ne 7.\]

Example

Try it.

For the following limit, define \(a,f(x),\) and \(L.\)

\[\underset{x\to 2}{\lim }\ (3x+5)=11\]

Solution

First, we recognize the notation of a limit. If the limit exists, as \(x\) approaches \(a,\) we write

\[\underset{x\to a}{\lim }\ f(x)=L.\]

We are given

\[\underset{x\to 2}{\lim }(3x+5)=11.\]

This means that \(a=2,f(x)=3x+5,\text{ and }L=11.\)

Condensed — the full section is in OpenStax Precalculus 2e.

Finding a Limit Using a Graph

To visually determine if a limit exists as \(x\) approaches \(a,\) we observe the graph of the function when \(x\) is very near to \(x=a.\) In we observe the behavior of the graph on both sides of \(a.\)

To determine if a left-hand limit exists, we observe the branch of the graph to the left of \(x=a,\) but near \(x=a.\) This is where \(x

To determine if a right-hand limit exists, observe the branch of the graph to the right of \(x=a,\) but near \(x=a.\) This is where \(x>a.\) We see that the outputs are getting close to some real number \(L,\) so there is a right-hand limit.

If the left-hand limit and the right-hand limit are the same, as they are in , then we know that the function has a two-sided limit. Normally, when we refer to a “limit,” we mean a two-sided limit, unless we call it a one-sided limit.

Finally, we can look for an output value for the function \(f(x)\) when the input value \(x\) is equal to \(a.\) The coordinate pair of the point would be \((a,f(a)).\) If such a point exists, then \(f(a)\) has a value. If the point does not exist, as in , then we say that \(f(a)\) does not exist.

Condensed — the full section is in OpenStax Precalculus 2e.

Finding a Limit Using a Table

Creating a table is a way to determine limits using numeric information. We create a table of values in which the input values of \(x\) approach \(a\) from both sides. Then we determine if the output values get closer and closer to some real value, the limit \(L.\)

Let’s consider an example using the following function:

\[\underset{x\to \ 5}{\lim }(\frac{{x}^{3}-125}{x-5})\]

To create the table, we evaluate the function at values close to \(x=5.\) We use some input values less than 5 and some values greater than 5 as in . The table values show that when \(x>5\) but nearing 5, the corresponding output gets close to 75. When \(x>5\) but nearing 5, the corresponding output also gets close to 75.

Because

\[\underset{x\to {5}^{-}}{\lim }f(x)=75=\underset{x\to {5}^{+}}{\lim }f(x),\]

then

\[\underset{x\to 5}{\lim }f(x)=75.\]

Remember that \(f(5)\) does not exist.

Example

Try it.

Numerically estimate the limit of the following expression by setting up a table of values on both sides of the limit.

\[\underset{x\to 0}{\lim }(\frac{5\sin (x)}{3x})\]

Solution

We can estimate the value of a limit, if it exists, by evaluating the function at values near \(x=0.\) We cannot find a function value for \(x=0\) directly because the result would have a denominator equal to 0, and thus would be undefined.

\[f(x)=\frac{5\sin (x)}{3x}\]

We create by choosing several input values close to \(x=0,\) with half of them less than \(x=0\) and half of them greater than \(x=0.\) Note that we need to be sure we are using radian mode. We evaluate the function at each input value to complete the table.

The table values indicate that when \(x<0\) but approaching 0, the corresponding output nears \(\frac{5}{3}.\)

When \(x>0\) but approaching 0, the corresponding output also nears \(\frac{5}{3}.\)

Because

\[\underset{x\to {0}^{-}}{\lim }f(x)=\frac{5}{3}=\underset{x\to {0}^{+}}{\lim }f(x),\]

then

\[\underset{x\to 0}{\lim }f(x)=\frac{5}{3}.\]

Condensed — the full section is in OpenStax Precalculus 2e.

Key Concepts

A function has a limit if the output values approach some value \(L\) as the input values approach some quantity \(a.\) See .
A shorthand notation is used to describe the limit of a function according to the form \(\underset{x\to \ a}{\lim }f(x)=L,\) which indicates that as \(x\) approaches \(a,\) both from the left of \(x=a\) and the right of \(x=a,\) the output value gets close to \(L.\)
A function has a left-hand limit if \(f(x)\) approaches \(L\) as \(x\) approaches \(a\) where \(xa.\)
A two-sided limit exists if the left-hand limit and the right-hand limit of a function are the same. A function is said to have a limit if it has a two-sided limit.
A graph provides a visual method of determining the limit of a function.
If the function has a limit as \(x\) approaches \(a,\) the branches of the graph will approach the same \(y\text{-}\) coordinate near \(x=a\) from the left and the right. See .
A table can be used to determine if a function has a limit. The table should show input values that approach \(a\) from both directions so that the resulting output values can be evaluated. If the output values approach some number, the function has a limit. See .
A graphing utility can also be used to find a limit. See .

Practice (40)

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

For the following limit, define \(a,f(x),\) and \(L.\)
\[\underset{x\to 2}{\lim }\ (3x+5)=11\]

Жауап беріңіз

First, we recognize the notation of a limit. If the limit exists, as \(x\) approaches \(a,\) we write
\[\underset{x\to a}{\lim }\ f(x)=L.\]
We are given
\[\underset{x\to 2}{\lim }(3x+5)=11.\]
This means that \(a=2,f(x)=3x+5,\text{ and }L=11.\)
For the following limit, define \(a,f(x),\) and \(L.\)
\[\underset{x\to 5}{\lim }(2{x}^{2}-4)=46\]

Жауап беріңіз

\(a=5,\) \(f(x)=2{x}^{2}-4,\) and \(L=46.\)
1. Determine the following limits and function value for the function \(f\) shown in .
  1. \(\underset{x\to {2}^{-}}{\lim }f(x)\)
  2. \(\underset{x\to {2}^{+}}{\lim }f(x)\)
  3. \(\underset{x\to 2}{\lim }f(x)\)
  4. \(f(2)\)
2. Determine the following limits and function value for the function \(f\) shown in .
  1. \(\underset{x\to {2}^{-}}{\lim }f(x)\)
  2. \(\underset{x\to {2}^{+}}{\lim }f(x)\)
  3. \(\underset{x\to 2}{\lim }f(x)\)
  4. \(f(2)\)
Жауап беріңіз
1. Looking at :
  \(\underset{x\to {2}^{-}}{\lim }f(x)=8;\) when \(x<2,\) but infinitesimally close to 2, the output values get close to \(y=8.\)
  \(\underset{x\to \ 2{\ }^{+}}{\lim }f(x)=3;\) when \(x>2,\) but infinitesimally close to 2, the output values approach \(y=3.\)
  \(\underset{x\to \ 2}{\lim }f(x)\) does not exist because \(\underset{x\to \ 2{\ }^{-}}{\lim }f(x)\ne \underset{x\to \ 2{\ }^{+}}{\lim }f(x);\) the left and right-hand limits are not equal.
  \(f(2)=3\) because the graph of the function \(f\) passes through the point \((2,f(2))\) or \((2,3).\)
2. Looking at :
  \(\underset{x\to \ 2{\ }^{-}}{\lim }f(x)=8;\) when \(x<2\) but infinitesimally close to 2, the output values approach \(y=8.\)
  \(\underset{x\to \ 2{\ }^{+}}{\lim }f(x)=8;\) when \(x>2\) but infinitesimally close to 2, the output values approach \(y=8.\)
  \(\underset{x\to \ 2}{\lim }f(x)=8\) because \(\underset{x\to \ 2{\ }^{-}}{\lim }f(x)=\underset{x\to \ 2{\ }^{+}}{\lim }f(x)=8;\) the left and right-hand limits are equal.
  \(f(2)=4\) because the graph of the function \(f\) passes through the point \((2,f(2))\) or \((2,4).\)
Using the graph of the function \(y=f(x)\) shown in , estimate the following limits.
1. \(\underset{x\to {0}^{-}}{\text{lim}}f(x)\)
2. \(\underset{x\to {0}^{+}}{\text{lim}}f(x)\)
3. \(\underset{x\to 0}{lim}f(x)\)
4. \(\underset{x\to {2}^{-}}{\text{lim}}f(x)\)
5. \(\underset{x\to {2}^{+}}{\text{lim}}f(x)\)
6. \(\underset{x\to 2}{lim}f(x)\)
7. \(\underset{x\to {4}^{-}}{\text{lim}}f(x)\)
8. \(\underset{x\to {4}^{+}}{\text{lim}}f(x)\)
9. \(\underset{x\to 4}{lim}f(x)\)
Жауап беріңіз

a. 0; b. 2; c. does not exist; d. \(-2;\) e. 0; f. does not exist; g. 4; h. 4; i. 4
Numerically estimate the limit of the following expression by setting up a table of values on both sides of the limit.
\[\underset{x\to 0}{\lim }(\frac{5\sin (x)}{3x})\]

Жауап беріңіз

We can estimate the value of a limit, if it exists, by evaluating the function at values near \(x=0.\) We cannot find a function value for \(x=0\) directly because the result would have a denominator equal to 0, and thus would be undefined.
\[f(x)=\frac{5\sin (x)}{3x}\]
We create by choosing several input values close to \(x=0,\) with half of them less than \(x=0\) and half of them greater than \(x=0.\) Note that we need to be sure we are using radian mode. We evaluate the function at each input value to complete the table.
The table values indicate that when \(x<0\) but approaching 0, the corresponding output nears \(\frac{5}{3}.\)
When \(x>0\) but approaching 0, the corresponding output also nears \(\frac{5}{3}.\)
Because
\[\underset{x\to {0}^{-}}{\lim }f(x)=\frac{5}{3}=\underset{x\to {0}^{+}}{\lim }f(x),\]
then
\[\underset{x\to 0}{\lim }f(x)=\frac{5}{3}.\]
Numerically estimate the limit of the following function by making a table:
\[\underset{x\to 0}{\lim }(\frac{20\sin (x)}{4x})\]

Жауап беріңіз

\(\underset{x\to 0}{\lim }(\frac{20\sin (x)}{4x})=5\)
With the use of a graphing utility, if possible, determine the left- and right-hand limits of the following function as \(x\) approaches 0. If the function has a limit as \(x\) approaches 0, state it. If not, discuss why there is no limit.
\[f(x)=3\sin (\frac{\pi }{x})\]

Жауап беріңіз

We can use a graphing utility to investigate the behavior of the graph close to \(x=0.\) Centering around \(x=0,\) we choose two viewing windows such that the second one is zoomed in closer to \(x=0\) than the first one. The result would resemble for \([-2,2]\) by \([-3,3].\)
The result would resemble for \([-0.1,0.1]\) by \([-3,3].\)
The closer we get to 0, the greater the swings in the output values are. That is not the behavior of a function with either a left-hand limit or a right-hand limit. And if there is no left-hand limit or right-hand limit, there certainly is no limit to the function \(f(x)\) as \(x\) approaches 0.
We write
\[\underset{x\to {0}^{-}}{\lim }(3\sin (\frac{\pi }{x}))\text{ does not exist}.\]\[\underset{x\to {0}^{+}}{\lim }(3\sin (\frac{\pi }{x}))\text{ does not exist}.\]\[\underset{x\to 0}{\lim }(3\sin (\frac{\pi }{x}))\text{ does not exist}\text{.}\]
Numerically estimate the following limit: \(\underset{x\to 0}{\lim }(\sin (\frac{2}{x})).\)

Жауап беріңіз

does not exist
Explain the difference between a value at \(x=a\) and the limit as \(x\) approaches \(a.\)

Жауап беріңіз

The value of the function, the output, at \(x=a\) is \(f(a).\) When the \(\underset{x\to a}{\lim }f(x)\) is taken, the values of \(x\) get infinitely close to \(a\) but never equal \(a.\) As the values of \(x\) approach \(a\) from the left and right, the limit is the value that the function is approaching.
Explain why we say a function does not have a limit as \(x\) approaches \(a\) if, as \(x\) approaches \(a,\) the left-hand limit is not equal to the right-hand limit.
\(\underset{x\to -{2}^{-}}{\lim }\ f(x)\)

Жауап беріңіз

–4
\(\underset{x\to -{2}^{+}}{\lim }\ f(x)\)
\(\underset{x\to -2}{\lim }\ f(x)\)

Жауап беріңіз

–4
\(\underset{x\to {1}^{-}}{\lim }\ f(x)\)

Жауап беріңіз

2
\(\underset{x\to {1}^{+}}{\lim }\ f(x)\)
\(\underset{x\to 1}{\lim }\ f(x)\)

Жауап беріңіз

does not exist
\(\underset{x\to {4}^{-}}{\lim }\ f(x)\)

Жауап беріңіз

4
\(\underset{x\to {4}^{+}}{\lim }\ f(x)\)
\(\underset{x\to 4}{\lim }\ f(x)\)

Жауап беріңіз

does not exist
\(\underset{x\to {0}^{-}}{\lim }\ f(x)=2\), \(\underset{x\to {0}^{+}}{\lim }\ f(x)=-3\), \(\underset{x\to 2}{\lim }\ f(x)=2\), \(\ f(0)=4\), \(\ f(2)=-1\), \(\ f(-3)\text{ does not exist}.\)

Жауап беріңіз

Answers will vary.
\(\underset{x\to {2}^{-}}{\lim }\ f(x)=0\), \(\ \underset{x\to {2}^{+}}{\lim }=-2\), \(\underset{x\to 0}{\lim }\ f(x)=3\), \(\ f(2)=5\), \(\ f(0)\)

Жауап беріңіз

Answers will vary.
\(\underset{x\to {2}^{-}}{\lim }\ f(x)=2\), \(\ \underset{x\to {2}^{+}}{\lim }\ f(x)=-3\), \(\ \underset{x\to 0}{\lim }\ f(x)=5\), \(\ f(0)=1\), \(\ f(1)=0\)

Жауап беріңіз

Answers will vary.
\(\underset{x\to {3}^{-}}{\lim }\ f(x)=0\), \(\ \underset{x\to {3}^{+}}{\lim }\ f(x)=5\), \(\ \underset{x\to 5}{\lim }\ f(x)=0\), \(\ f(5)=4\), \(\ f(3)\text{ does not exist}.\)

Жауап беріңіз

Answers will vary.
\(\underset{x\to 4}{\lim }\ f(x)=6\), \(\ \underset{x\to {6}^{+}}{\lim }\ f(x)=-1\), \(\ \underset{x\to 0}{\lim }\ f(x)=5\), \(\ f(4)=6\), \(\ f(2)=6\)

Жауап беріңіз

Answers will vary.
\(\underset{x\to -3}{\lim }\ f(x)=2\), \(\ \underset{x\to {1}^{+}}{\lim }\ f(x)=-2\), \(\ \underset{x\to 3}{\lim }\ f(x)=-4\), \(\ f(-3)=0\), \(\ f(0)=0\)

Жауап беріңіз

Answers will vary.
\(\underset{x\to \pi }{\lim }\ f(x)={\pi }^{2}\), \(\ \underset{x\to -\pi }{\lim }\ f(x)=\frac{\pi }{2}\), \(\ \underset{x\to {1}^{-}}{\lim }\ f(x)=0\), \(\ f(\pi )=\sqrt{2}\), \(\ f(0)\text{ does not exist}.\)

Жауап беріңіз

Answers will vary.
\(f(x)={(1+x)}^{\frac{1}{x}}\)
\(g(x)={(1+x)}^{\frac{2}{x}}\)

Жауап беріңіз

7.38906
\(h(x)={(1+x)}^{\frac{3}{x}}\)
\(i(x)={(1+x)}^{\frac{4}{x}}\)

Жауап беріңіз

54.59815
\(j(x)={(1+x)}^{\frac{5}{x}}\)
Based on the pattern you observed in the exercises above, make a conjecture as to the limit of \(f(x)={(1+x)}^{\frac{6}{x}},\) \(g(x)={(1+x)}^{\frac{7}{x}},\) \(\text{and }h(x)={(1+x)}^{\frac{n}{x}}.\)

Жауап беріңіз

\({e}^{6}\approx 403.428794,\) \({e}^{7}\approx 1096.633158,\) \({e}^{n}\)
\((x)=\{\begin{array}{ll}|x|-1, & \text{if }x\ne 1 \\ {x}^{3}, & \text{if }x=1\end{array}\ a=1\)
\((x)=\{\begin{array}{ll}\frac{1}{x+1}, & \text{if }x=-2 \\ {(x+1)}^{2}, & \text{if }x\ne -2\end{array}\ a=-2\)

Жауап беріңіз

\(\underset{x\to -2}{\lim }f(x)=1\)
\(f(x)=\frac{{x}^{2}-4x}{16-{x}^{2}};a=4\)
\(f(x)=\frac{{x}^{2}-x-6}{{x}^{2}-9};a=3\)

Жауап беріңіз

\(\underset{x\to 3}{\lim }(\frac{{x}^{2}-x-6}{{x}^{2}-9})=\frac{5}{6}\approx 0.83\)
\(f(x)=\frac{{x}^{2}-6x-7}{{x}^{2}-\ 7x};a=7\)
\(f(x)=\frac{{x}^{2}-1}{{x}^{2}-3x+2};a=1\)

Жауап беріңіз

\(\underset{x\to 1}{\lim }(\frac{{x}^{2}-1}{{x}^{2}-3x+2})=-2.00\)
\(f(x)=\frac{1-{x}^{2}}{{x}^{2}-3x+2};a=1\)
\(f(x)=\frac{10-10{x}^{2}}{{x}^{2}-3x+2};a=1\)

Жауап беріңіз

\(\underset{x\to 1}{\lim }(\frac{10-10{x}^{2}}{{x}^{2}-3x+2})=20.00\)

Symbols used here

\lim_{x \to a} f(x)

limit
The value f(x) approaches as x approaches a.

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

\sin,\ \cos,\ \tan

sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.

imaginary unit
i² = −1.

\neq

not equal
The two sides are different.

Euler's number
2.71828…, the base whose exponential is its own derivative.

\infty

infinity
Not a number: "grows without bound" in limits and intervals.

\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.

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: Finding Limits: Numerical and Graphical Approaches

Understand limit notation.
Find a limit using a graph.
Find a limit using a table.
Examine the graph to determine whether a left-hand limit exists.
Examine the graph to determine whether a right-hand limit exists.
If the two one-sided limits exist and are equal, then there is a two-sided limit—what we normally call a “limit.”
If there is a point at
Determine the following limits and function value for the function

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.

Өзіңіздіңіңізді сынап көріңіз

Parts of this page are adapted from OpenStax Precalculus 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Келесіде Calculus

Limits Derivatives Integrals Definite integrals Taylor series Series and sums Maxima and minima The chain rule Implicit differentiation Related rates and optimisation Integration techniques: substitution, parts, partial fractions Applications of integration: area, volume, arc length Infinite series and convergence tests