Lecture · 16 min · intermediate

Limits

Where a function is heading, even if it never arrives

Courant · limits; Hardy · continuous functions

The difference quotient $\frac{f ( a + h ) - f ( a )}{h}$ is undefined at $h = 0$ . Calculus begins with a refusal to plug in $h = 0$ , and a decision to ask instead: what number does this approach as $h$ gets arbitrarily close to $0$ ? That number is the limit. Differentiability is a special case of this question.

On the board today

—Read $lim_{x \to a} f (x) = L$ in ordinary language.
—Compute simple algebraic limits by rewriting.
—Distinguish a limit from a function value, including removable holes.

Informal definition

x \to a lim f (x) = L

$f (x)$ can be made as close to $L$ as we like, by taking $x$ sufficiently close to $a$ (but not equal to $a$ ).

The function need not be defined at $a$ . The classic $\frac{s i n x}{x}$ has no value at $0$ , but the limit is $1$ . Continuity at $a$ is the extra demand that $f (a)$ exists and equals the limit. A removable discontinuity is a hole you could fill with the limiting value.

Worked example · A hole you can fill

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

1.The expression is $0/0$ if you substitute — undefined, not “zero.”
2.Factor: $\frac{( x - 2 ) ( x + 2 )}{x - 2} = x + 2$ for $x \neq = 2$ .
$x \to 2 lim (x + 2) = 4$

The limit is $4$ . The original function has a hole at $x = 2$ .

Pause the lecture

Does $lim_{x \to 0} \frac{∣ x ∣}{x}$ exist?

Common mistake. Substituting and getting $0/0$ , then writing “undefined, so the limit DNE.” $0/0$ is an invitation to rewrite, not a verdict. Limits are about nearby values, not the missing point.

Practice this lecture

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