Limits and Continuity

Definition of a Limit

The limit of $f(x)$ as $x$ approaches $a$ is $L$, written as:

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

This means $f(x)$ gets arbitrarily close to $L$ as $x$ gets close to $a$.

One-Sided Limits

Left-Hand Limit

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

Right-Hand Limit

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

The limit exists if and only if both one-sided limits exist and are equal.

Limit Laws

Let $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$. Then:

Law	Formula
Sum	$\lim [f(x) + g(x)] = L + M$
Difference	$\lim [f(x) - g(x)] = L - M$
Product	$\lim [f(x) \cdot g(x)] = L \cdot M$
Quotient	$\lim [f(x)/g(x)] = L/M$, if $M \neq 0$
Power	$\lim [f(x)]^n = L^n$
Constant	$\lim c = c$
Identity	$\lim x = a$

Important Limits

$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$

$\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$

$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$

$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$

Continuity

A function $f$ is continuous at $x = a$ if:

1. $f(a)$ is defined
2. $\lim_{x \to a} f(x)$ exists
3. $\lim_{x \to a} f(x) = f(a)$

Types of Discontinuities

• Removable: limit exists but $f(a)$ is undefined or $\neq$ limit
• Jump: left and right limits exist but are different
• Infinite: function approaches $\pm\infty$

Intermediate Value Theorem

If $f$ is continuous on $[a, b]$ and $k$ is between $f(a)$ and $f(b)$, then there exists at least one $c$ in $(a, b)$ such that $f(c) = k$.

Examples

Example 1: Evaluate the limit

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

$= \lim_{x \to 2} \frac{(x + 2)(x - 2)}{x - 2}$

$= \lim_{x \to 2} (x + 2) = 4$

Example 2: One-sided limits

For $f(x) = \frac{|x|}{x}$:

• $\lim_{x \to 0^-} f(x) = -1$ (from the left)
• $\lim_{x \to 0^+} f(x) = 1$ (from the right)
• Limit does not exist at $x = 0$