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Limit of (x^2-1)/(x-1) as x->1: animated lesson

A narrated, step-by-step limits video lesson with captions. Final answer: Limit = 2.

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Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.7 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
$\lim_{x \to 1} \frac{{x}^{2}-1}{x-1}$
2. Step by step
1. $\lim_{x \to 1} \frac{{x}^{2}-1}{x-1} = \lim_{x \to 1} \frac{2x}{1}$
  Indeterminate form 0/0 — apply L'Hôpital's rule
2. $\lim_{x \to 1} \frac{2x}{1} = \lim_{x \to 1} 2x$
  Simplify
3. $\lim_{x \to 1} 2x = 2\cdot1 = 2$
  Direct substitution
3. The answer
$\lim_{x \to 1} \frac{{x}^{2}-1}{x-1} = 2$
lim x→1 (x² - 1) / (x - 1) = 2
4. Check
- Numerical values from the right agree: f(1.01) = 2.01, f(1.001) = 2.001, f(1.0001) = 2.0001, f(1.00001) = 2.00001 ✓
- Numerical values from the left agree: f(0.99) = 1.99, f(0.999) = 1.999, f(0.9999) = 1.9999, f(0.99999) = 1.99999 ✓

Transcript

The problem. Here is the problem: (x squared minus 1)/(x minus 1). Find the limit of (x squared minus 1) / (x minus 1) as x → 1.

Step by step. Indeterminate form 0 over 0 — apply L prime Hôpital prime s rule. Differentiate the numerator and the denominator separately (this is not the quotient rule). Simplify. Direct substitution. The expression is continuous at x equals 1, so the limit equals its value there.

The answer. So the answer is Limit equals 2. A limit describes the value a function approaches, not necessarily the value it takes. When substitution gives an indeterminate form, the expression is rewritten (L prime Hôpital prime s rule, a conjugate, or logarithms) until substitution works.

Check. Numerical values from the right agree: f(1.01) equals 2.01, f(1.001) equals 2.001, f(1.0001) equals 2.0001, f(1.00001) equals 2.00001, which checks out. Numerical values from the left agree: f(0.99) equals 1.99, f(0.999) equals 1.999, f(0.9999) equals 1.9999, f(0.99999) equals 1.99999, which checks out.

About limits

Evaluate limits by substitution, factoring, standard limits and L'Hôpital's rule.

Limit of (x^2-1)/(x-1) as x->1: Limits Step-by-Step Lesson