Limits & Continuity Cheat Sheet

A printable reference covering limit laws, one-sided limits, infinite limits, continuity, and the Intermediate Value Theorem for grades 11-12.

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Limits describe the value a function approaches as the input gets close to a number, even when the function is not defined there. This cheat sheet helps students organize the main rules for evaluating limits quickly and accurately. It also connects limits to continuity, which is one of the foundations of calculus. Students need these ideas before learning derivatives, tangent lines, and many real-world rate problems. Core concepts include direct substitution, algebraic simplification, one-sided limits, infinite limits, and special limit theorems. A function is continuous at $x=a$ when $f(a)$ exists, $\lim_{x \to a} f(x)$ exists, and $\lim_{x \to a} f(x)=f(a)$ . Important formulas include the quotient law $\lim_{x \to a} \frac{f(x)}{g(x)}=\frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$ when the denominator limit is not $0$ . The Intermediate Value Theorem explains why continuous functions take every value between two endpoint values on a closed interval.

Key Facts

Direct substitution works when $f$ is continuous at $x=a$ , so $\lim_{x \to a} f(x)=f(a)$ .
The sum and difference laws state that $\lim_{x \to a} [f(x) \pm g(x)]=\lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$ when both limits exist.
The product law states that $\lim_{x \to a} [f(x)g(x)]=[\lim_{x \to a} f(x)][\lim_{x \to a} g(x)]$ when both limits exist.
The quotient law states that $\lim_{x \to a} \frac{f(x)}{g(x)}=\frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$ when $\lim_{x \to a} g(x) \ne 0$ .
A two-sided limit exists only if $\lim_{x \to a^-} f(x)=\lim_{x \to a^+} f(x)$ .
A vertical asymptote occurs at $x=a$ if $\lim_{x \to a} f(x)=\infty$ or $\lim_{x \to a} f(x)=-\infty$ .
The Squeeze Theorem says if $g(x) \le f(x) \le h(x)$ near $x=a$ and $\lim_{x \to a} g(x)=\lim_{x \to a} h(x)=L$ , then $\lim_{x \to a} f(x)=L$ .
Continuity at $x=a$ requires $f(a)$ to exist, $\lim_{x \to a} f(x)$ to exist, and $\lim_{x \to a} f(x)=f(a)$ .

Vocabulary

Limit: A limit is the value that $f(x)$ approaches as $x$ gets close to a number $a$ .
One-sided limit: A one-sided limit describes what $f(x)$ approaches as $x$ approaches $a$ from only the left or only the right.
Continuity: Continuity at $x=a$ means the graph has no hole, jump, or break there, and $\lim_{x \to a} f(x)=f(a)$ .
Removable discontinuity: A removable discontinuity is a hole in the graph where the limit exists but the function value is missing or different.
Jump discontinuity: A jump discontinuity occurs when $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ both exist but are not equal.
Vertical asymptote: A vertical asymptote is a line $x=a$ where the function grows without bound toward $\infty$ or $-\infty$ .

Common Mistakes to Avoid

Substituting immediately into every limit is wrong because expressions such as $\frac{0}{0}$ are indeterminate and need simplification first.
Assuming a hole changes the limit is wrong because a limit depends on nearby values of $f(x)$ , not only on the value of $f(a)$ .
Combining one-sided limits without checking equality is wrong because $\lim_{x \to a} f(x)$ exists only when $\lim_{x \to a^-} f(x)=\lim_{x \to a^+} f(x)$ .
Treating $\frac{k}{0}$ as $0$ is wrong because division by $0$ is undefined, and the expression may indicate an infinite limit or no limit.
Calling a function continuous just because it is defined at $x=a$ is wrong because continuity also requires $\lim_{x \to a} f(x)$ to exist and equal $f(a)$ .

Practice Questions

1 Evaluate $\lim_{x \to 3} \frac{x^2-9}{x-3}$ .
2 Evaluate $\lim_{x \to 0} \frac{\sin x}{x}$ .
3 For $f(x)=\begin{cases}2x+1, & x<1 \\ 5, & x=1 \\ x^2+2, & x>1\end{cases}$ , determine whether $\lim_{x \to 1} f(x)$ exists and whether $f$ is continuous at $x=1$ .
4 Explain why a function can have a limit at $x=a$ even if it is not continuous at $x=a$ .