Continuity and limits: complete lesson (definitions, rules, indeterminates)

Limit at a point

$\lim_{x \to a} f(x) = \ell$ means $f(x)$ “approaches” $\ell$ as $x$ “approaches” $a$.

• $\ell$ finite: finite limit.
• $\ell = \pm\infty$: infinite limit.

Limit at infinity

$\lim_{x \to +\infty} f(x) = \ell$: $f(x)$ approaches $\ell$ as $x$ grows large.

Same for $x \to -\infty$.

Reference limits

$x \to$	$x^n$	$1/x^n$	$\sqrt{x}$	$e^x$	$\ln x$
$+\infty$	$+\infty$	$0$	$+\infty$	$+\infty$	$+\infty$
$0^+$	$0$	$+\infty$	$0$	$1$	$-\infty$

Algebra of limits

Assuming both sides exist:

$\lim (f + g) = \lim f + \lim g$ $\lim (f \times g) = \lim f \times \lim g$ $\lim (f / g) = \lim f / \lim g \quad (\lim g \neq 0)$

⚠️ Watch out for indeterminate forms:

• $\infty - \infty$
• $0 \times \infty$
• $0 / 0$
• $\infty / \infty$
• $1^\infty$
• $0^0$

Resolving indeterminate forms

$\infty / \infty$ for a rational function: keep the dominant terms.

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

$0/0$: factor and simplify.

$\lim_{x \to 2} \dfrac{x^2 - 4}{x - 2} = \lim_{x \to 2} (x + 2) = 4$.

Growth comparisons:

• $\lim_{x \to +\infty} \dfrac{e^x}{x^n} = +\infty$ (exp beats polynomial).
• $\lim_{x \to +\infty} \dfrac{\ln x}{x^\alpha} = 0$ for $\alpha > 0$ (polynomial beats log).

Asymptotes

• Horizontal asymptote at $+\infty$ if $\lim_{x \to +\infty} f = \ell$ (finite): $y = \ell$.
• Vertical asymptote at $a$ if $\lim_{x \to a} f = \pm\infty$: $x = a$.
• Oblique asymptote at $+\infty$ if $\lim [f(x) - (ax + b)] = 0$: $y = ax + b$.

Continuity

$f$ is continuous at $a$ if $\lim_{x \to a} f(x) = f(a)$.

$f$ is continuous on $I$ if continuous at every point of $I$.

Theorem: polynomials, exp, ln, sin, cos, $\sqrt{}$, quotients (where the denominator doesn’t vanish) are continuous on their domain.

Intermediate Value Theorem (IVT)

If $f$ is continuous on $[a, b]$ and $k$ lies between $f(a)$ and $f(b)$, there exists $c \in [a, b]$ with $f(c) = k$.

Bijection corollary: if additionally $f$ is strictly monotonic, then $c$ is unique.

Common mistakes

1. Taking the limit of a quotient without checking the denominator.
2. Using $\infty - \infty = 0$ (false: indeterminate).
3. Forgetting continuity before applying IVT.
4. Confusing limit at a point and value at that point.