CalculusCalculus 1College

Limits and Continuity

Evaluate limits algebraically and decide continuity.

A limit describes the value a function approaches as its input approaches some number, whether or not the function is defined there. Writing $\lim_{x \to a} f(x) = L$ means $f(x)$ can be made as close to $L$ as you like by taking $x$ close enough to $a$ , from both sides.

Limits are the foundation of calculus: derivatives are limits of difference quotients and integrals are limits of sums. The practical toolkit is short: try direct substitution, and if that gives the indeterminate form $\frac{0}{0}$ , simplify by factoring, rationalising or using a known limit, then substitute again.

Key terms

Limit: The value f(x) approaches as x approaches a.
$\lim_{x\to a} f(x) = L$

Key ideas

If $f$ is continuous at $a$ , the limit is simply $f(a)$ .
A two-sided limit exists only if the left and right limits agree.
$\lim_{x \to 0} \frac{\sin x}{x} = 1$ with $x$ in radians.
For rational functions at infinity, compare the degrees of numerator and denominator.

Worked example

Evaluate $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$ .

Substitution gives $\frac{0}{0}$ , so simplify.
Factor: $\frac{(x - 3)(x + 3)}{x - 3}$ .
Cancel for $x \ne 3$ : $x + 3$ .
Substitute: $3 + 3 = 6$ .

Answer: $6$

Common mistakes

Declaring a $\frac{0}{0}$ limit to be 0 or undefined without simplifying.
Assuming the limit equals the function value at a hole.
Checking only one side for piecewise functions.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.Evaluate $\lim_{x \to 2} (3x^2 - 1)$ .
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$11$
2.Evaluate $\lim_{x \to \infty} \frac{4x^2 + 1}{2x^2 - 5}$ .
Show answer
$2$
3.Evaluate $\lim_{x \to 1} \frac{x^2 + x - 2}{x - 1}$ .
Show answer
$3$

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