Limits

What is a limit?

The notation

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

means: as $x$ gets arbitrarily close to $a$, $f(x)$ gets arbitrarily close to $L$. Importantly, it does not require that $f(a) = L$ — $f$ doesn't even have to be defined at $a$.

Method 1: Just plug it in

If $f$ is continuous at $a$ (no holes, jumps, or vertical asymptotes), then:

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

Example. $\lim_{x \to 2} (x^2 + 3) = 4 + 3 = 7$. Done.

Method 2: Algebra for indeterminate forms

If plugging in gives $\frac{0}{0}$, simplify first.

Example. $\lim_{x \to 3} \dfrac{x^2 - 9}{x - 3}$.

Direct substitution gives $\frac{0}{0}$ — bad. Factor:

$\frac{x^2 - 9}{x - 3} = \frac{(x-3)(x+3)}{x - 3} = x + 3$

Now plug in: $3 + 3 = 6$.

One-sided limits

$\lim_{x \to a^-}$ means approach from the left (smaller values). $\lim_{x \to a^+}$ means approach from the right (larger values).

The full (two-sided) limit exists only if both sides agree:

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

Limits at infinity

$\lim_{x \to \infty}$ asks: what happens as $x$ grows without bound?

Example. $\lim_{x \to \infty} \dfrac{3x^2 + 1}{x^2 - 5}$.

Divide top and bottom by $x^2$: $\dfrac{3 + 1/x^2}{1 - 5/x^2}$. As $x \to \infty$, the $1/x^2$ terms go to $0$, leaving $3/1 = 3$.

Shortcut for rational functions: compare the degrees of the numerator and denominator.

• Numerator degree < denominator degree → limit is $0$.
• Numerator degree = denominator degree → limit is the ratio of leading coefficients.
• Numerator degree > denominator degree → limit is $\pm \infty$.

Key takeaways

• Limits describe behavior near a point, not necessarily at it.
• $\frac{0}{0}$ means "try harder, don't conclude." Usually factor and cancel.
• For one-sided behavior, check the left and right separately.