Evaluating Limits

A limit tells you the value a function approaches as x gets close to a point, even when the function is not defined at that point. Here are the five methods you will use most.

Direct substitution
Factorization
Rationalization
Standard trigonometric limits
One-sided limits

Which method first?

Substitute the value. If you get a number, you are done.
If you get \tfrac{0}{0}, factor (polynomials) or rationalize (square roots), or recognise a standard trig limit.
If the two sides might disagree, check each side separately.

Direct substitution

Use it when the function is defined at the point. Polynomials, sines, cosines and exponentials are smooth everywhere, so the limit is just the value: nothing goes wrong as you slide in.

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

Example. Find \lim_{x\to 2}(x^2+3x).

Put x=2 in: 2^2+3(2)=4+6=10. The limit is 10.

Watch out: if you land on \tfrac{0}{0}, the limit may still exist. That is an indeterminate form: switch to factorization, rationalization or a standard limit. If you get a non-zero number over zero, the values grow without bound.

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Factorization

Use it when a fraction gives \tfrac{0}{0}. A zero on top and bottom means both share a factor (x-a). Because a limit only asks about nearby x, never x = a itself, you can cancel it.

\frac{x^2-1}{x-1}=\frac{(x-1)(x+1)}{x-1}=x+1\quad(x\ne1)

Example. Find \lim_{x\to1}\dfrac{x^2-1}{x-1}.

Substituting gives \tfrac{0}{0}.
Factor the top: x^2-1=(x-1)(x+1).
Cancel x-1 to leave x+1, then substitute: 1+1=2.

The limit is 2.

Watch out: cancel factors only, never terms. In \tfrac{x+3}{x} you cannot cancel the x.

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Rationalization

Use it when a square root causes \tfrac{0}{0}. Multiply top and bottom by the conjugate (flip the sign between the two terms). The difference of squares removes the root and exposes a cancelling factor.

(\sqrt{x}-2)(\sqrt{x}+2)=x-4

Example. Find \lim_{x\to4}\dfrac{\sqrt{x}-2}{x-4}.

Substituting gives \tfrac{0}{0}.
Multiply by \tfrac{\sqrt{x}+2}{\sqrt{x}+2}. The top becomes x-4.
Cancel: \dfrac{x-4}{(x-4)(\sqrt{x}+2)}=\dfrac{1}{\sqrt{x}+2}. Substitute: \tfrac{1}{4}.

The limit is 1/4.

Watch out: multiply the top and the bottom by the same conjugate, so the fraction keeps its value.

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Standard trigonometric limits

Use it when a sine, cosine or tangent is compared with x as x approaches 0. Substitution gives \tfrac{0}{0}, but these results are known facts you can reuse. They hold when the angle is in radians.

\lim_{x\to0}\frac{\sin x}{x}=1\qquad\lim_{x\to0}\frac{\tan x}{x}=1\qquad\lim_{x\to0}\frac{1-\cos x}{x^2}=\frac12

Example. Find \lim_{x\to0}\dfrac{\sin 3x}{x}.

Match the angle: \dfrac{\sin 3x}{x}=3\cdot\dfrac{\sin 3x}{3x}. As x\to0 the fraction tends to 1, so the limit is 3.

Watch out: the angle inside sine must match what is underneath. You may need to multiply and divide by a constant to line them up.

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One-sided limits

Use it when the function behaves differently on the two sides of a point, or blows up there. Typical cases are absolute values, piecewise functions and fractions with a zero denominator. Write a^- for "from the left" and a^+ for "from the right".

\lim_{x\to a}f(x)\text{ exists}\iff\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)

Example. Find \lim_{x\to0}\dfrac{|x|}{x}.

From the right, x>0 so \tfrac{|x|}{x}=1.
From the left, x<0 so \tfrac{|x|}{x}=-1.

The sides disagree, so the two-sided limit does not exist.

Watch out: a one-sided limit can exist even when the two-sided limit does not. In the calculator, choose From the left or From the right to check each side.

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Where limits are used

Limits define the derivative and the definite integral, and they decide whether a function is continuous. Every rule in the other guides is built on them.

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