Limits

Substitution, factoring and L'Hôpital's rule.

A limit asks what a function approaches, not what it equals. Try substitution first; if it gives 0/0, factor and cancel, or differentiate top and bottom (L'Hôpital). The graph shows the function near the point with the limit marked.

Àwọn ààtò àwọn ìṣàmúlò-ètò · limit of sin(x)/x as x -> 0

Calculus · Limits

Limit of sin(x)/x as x → 0

\lim_{x \to 0} \frac{\sin{\left(x \right)}}{x}

Àwọn ìṣàmúlò-ètò

\lim_{x \to 0^+-} \frac{\sin{\left(x \right)}}{x}
Try direct substitution first.
\frac{0}{0}
Substitution gives 0/0: an indeterminate form.
\lim \frac{\cos{\left(x \right)}}{1}
L'Hôpital's rule: differentiate numerator and denominator separately.
= 1
Substitute.

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Bii o ṣe le: Limits

Try direct substitution first.
Substitution gives 0/0: an indeterminate form.
L'Hôpital's rule: differentiate numerator and denominator separately.
Substitute.

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Calculus

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