Read the notation
\(\lim_{x\to a}f(x)=L\)
\(x\) approaches the input \(a\); the function values approach the output \(L\).
Tahat Just Math · Self-paced practice
Understand and Practice Limits
Evaluate limits from formulas, tables, and graphs; simplify radicals; and analyze infinite limits.
- Evaluate limits using direct substitution.
- Apply the limit laws for sums, differences, products, and quotients.
- Resolve indeterminate forms by algebraic manipulation.
- Read limits from tables and graphs and analyze one-sided limits.
- Simplify radical limits using conjugates.
- Evaluate infinite limits and identify vertical asymptotes.
- Apply the Squeeze Theorem.
- Evaluate trigonometric limits using fundamental limits and identities.
50 questions · 10 learning objectives
Core idea
A limit describes nearby values
\(\lim_{x\to a}f(x)=L\) means that \(f(x)\) approaches \(L\) as \(x\) approaches \(a\). The value of \(f(a)\) may be different from \(L\), or may not be defined at all.
Check from both sides
A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
\(\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L\)
Evaluate
Use direct substitution and limit laws
For polynomials and functions continuous at \(a\), substitute \(x=a\). Limits also distribute across sums, differences, constant multiples, products, and quotients when the denominator's limit is nonzero.
Direct substitution
\(\begin{aligned}&\lim_{x\to 2}(x^2+3x-1)=2^2+3(2)-1\\&=9\end{aligned}\)
Substitution gives the limit immediately when the function is continuous at the target.
Limit laws
\(\begin{aligned}&\lim_{x\to a}(f(x)+g(x))=\lim_{x\to a}f(x)+\lim_{x\to a}g(x)\end{aligned}\)
\(\begin{aligned}&\lim_{x\to a}(f(x)g(x))=\left(\lim_{x\to a}f(x)\right)\left(\lim_{x\to a}g(x)\right)\end{aligned}\)
Evaluate simpler pieces first, then combine their limits using the matching law.
Trigonometric limits
Use fundamental limits and identities
Use radians for these limits. First try direct substitution: sine and cosine are continuous everywhere, and tangent and reciprocal functions are continuous wherever defined. If substitution gives \(\frac{0}{0}\), use an identity to rewrite the expression.
Fundamental limits
\(\lim_{x\to0}\frac{\sin x}{x}=1\)
\(\lim_{x\to0}\frac{\cos x-1}{x}=0\)
\(\lim_{x\to0}\frac{\tan x}{x}=1\)
These formulas require radian measure. For a scaled angle, rewrite the ratio so the angle and denominator match.
Reciprocal and Pythagorean identities
\(\csc x=\frac{1}{\sin x},\quad \sec x=\frac{1}{\cos x}\)
\(\cot x=\frac{1}{\tan x},\quad \sin^2x+\cos^2x=1\)
\(1+\tan^2x=\sec^2x,\quad 1+\cot^2x=\csc^2x\)
Use the identities where both sides are defined; for example, \(1-\cos x=\frac{\sin^2x}{1+\cos x}\) near zero.
Addition and subtraction formulas
\(\sin(a\pm b)=\sin a\cos b\pm\cos a\sin b\)
\(\cos(a\pm b)=\cos a\cos b\mp\sin a\sin b\)
\(\tan(a\pm b)=\frac{\tan a\pm\tan b}{1\mp\tan a\tan b}\)
Expand a shifted angle to separate the part approaching zero.
Double-angle formulas
\(\sin(2x)=2\sin x\cos x\)
\(\cos(2x)=1-2\sin^2x=2\cos^2x-1\)
\(\tan(2x)=\frac{2\tan x}{1-\tan^2x}\)
Apply the identity only where its denominator is nonzero.
Indeterminate form
Factor, simplify, then substitute
If direct substitution gives \(\frac{0}{0}\), the expression is indeterminate; that result is not the limit. Factor and simplify for \(x\ne a\), then evaluate the simplified expression as \(x\to a\).
Worked algebraic structure
\(\begin{aligned}&\lim_{x\to 2}\frac{x^2-4}{x-2}=\lim_{x\to 2}(x+2)\\&=4\end{aligned}\)
The original function is undefined at \(x=2\), but its limit exists because nearby values approach \(4\).
Tables and graphs
Estimate limits from nearby values
Read values as \(x\) approaches the target from both sides. A table suggests an estimate, not a proof; a graph shows the approached height, not necessarily the filled point at the target.
Read both sides
Near \(x=2\), values \(2.9601\) and \(3.0401\) suggest that both sides approach \(3\).
\(\begin{aligned}&\lim_{x\to2}f(x)\approx3\end{aligned}\)
Infinite limits
Recognize vertical asymptotes
If a function grows without bound on at least one side as \(x\to a\), the line \(x=a\) is a vertical asymptote. Check the signs from each side; infinity is not a finite limit.
A positive squared denominator
\(\begin{aligned}&\lim_{x\to1}\frac{4}{(x-1)^2}=+\infty\end{aligned}\)
The denominator approaches zero through positive values on both sides, so \(x=1\) is a vertical asymptote.
Squeeze Theorem
Find a limit using bounds
When \(g(x)\le f(x)\le h(x)\) near \(a\), and both outer functions approach the same limit \(L\), the function between them also approaches \(L\).
An oscillating function
\(\begin{aligned}&-x^2\le x^2\sin(1/x)\le x^2\\&\lim_{x\to0}x^2\sin(1/x)=0\end{aligned}\)
Both bounds approach \(0\), even though \(\sin(1/x)\) oscillates.
Limit Lab
Type the function or tap the math keys. The cursor shows where the next symbol will go.
Learn by doing
Guided examples by skill
Open a skill, read the idea, then work through one example a step at a time. Use each question’s hint and step-by-step solution controls in Practice problems as needed.
Chapter 1 · Mock exam
A focused 20-question test
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Random practice
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Chapter 1 contains 50 questions across 10 learning objectives. Choose any number of questions you like. A random set takes one question from each objective first, so a set of 8 or more questions covers the whole chapter.