CALCULUS · LIMITS

Understanding Limits

A limit describes where a function is heading, not necessarily where it is.

14 min read 3 practice questions

The limit

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

means that as

x

gets closer and closer to

a

(from both sides),

f(x)

gets closer and closer to

L

. The value

f(a)

itself doesn't matter — it might not even exist.

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

A two-sided limit exists only when both one-sided limits agree

Worked example · Factor and cancel

Evaluate

\displaystyle\lim_{x \to 3} \frac{x^2 - 9}{x - 3}

1
Direct substitution gives $\frac{0}{0}$ — an indeterminate form, so we need algebra.
2
Factor: $\frac{(x - 3)(x + 3)}{x - 3} = x + 3$ for $x \neq 3$ .
3
Now substitute: $\lim_{x \to 3} (x + 3) = 6$ .

Two special limits show up again and again in calculus:

\lim_{x \to 0} \frac{\sin x}{x} = 1 \qquad\qquad \lim_{x \to \infty} \frac{1}{x} = 0

CHECK YOUR UNDERSTANDING

Practice

Answer each question, then check.

Evaluate

\displaystyle\lim_{x \to 2} (3x^2 - 1)

Evaluate

\displaystyle\lim_{x \to 5} \frac{x^2 - 25}{x - 5}

Evaluate

\displaystyle\lim_{x \to \infty} \frac{4x^2 + 1}{2x^2 - 3}

Numeric answers accept decimals or fractions like 3/4.