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Guides AP and college calculus

How to Find Limits Algebraically

Start by substituting. If you get 0/0, you still have work to do.

Substitute the value.
If you get $\frac{0}{0}$ , simplify and substitute again.
For limits at infinity, compare the highest powers.

Worked examples

Find $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$

Substitution gives $\frac{0}{0}$ .

$\frac{( x - 2 ) ( x + 2 )}{x - 2} = x + 2$ when $x \neq = 2$

The limit is $4$ .

Find $lim_{x \to \infty} \frac{3 x ^{2} + 1}{x ^{2} - 5}$

The top and bottom have the same highest power.

Compare leading coefficients: the limit is $3$ .

Common mistake

Saying a limit doesn't exist because you got $\frac{0}{0}$ .

$\frac{0}{0}$ is an indeterminate form. It means you need another method before deciding.

Try it yourself

$lim_{x \to 3} \frac{x ^{2} - 9}{x - 3}$
Show answer
$6$
$lim_{x \to 1} \frac{x ^{2} + x - 2}{x - 1}$
Show answer
$3$
$lim_{x \to \infty} \frac{2 x + 1}{x ^{2}}$
Show answer
$0$

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