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Find lim x->3 (x^2-9)/(x-3): animated lesson

A narrated, step-by-step limits video lesson with captions. Final answer: Limit = 6.

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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
$\lim_{x \to 3} \frac{{x}^{2}-9}{x-3}$
2. Step by step
1. $\lim_{x \to 3} \frac{{x}^{2}-9}{x-3} = \lim_{x \to 3} \frac{2x}{1}$
  Indeterminate form 0/0 — apply L'Hôpital's rule
2. $\lim_{x \to 3} \frac{2x}{1} = \lim_{x \to 3} 2x$
  Simplify
3. $\lim_{x \to 3} 2x = 2\cdot3 = 6$
  Direct substitution
3. The answer
$\lim_{x \to 3} \frac{{x}^{2}-9}{x-3} = 6$
lim x→3 (x² - 9) / (x - 3) = 6
4. Check
- Numerical values from the right agree: f(3.01) = 6.01, f(3.001) = 6.001, f(3.0001) = 6.0001, f(3.00001) = 6.00001 ✓
- Numerical values from the left agree: f(2.99) = 5.99, f(2.999) = 5.999, f(2.9999) = 5.9999, f(2.99999) = 5.99999 ✓

Transcript

The problem. Here is the problem: (x squared minus 9)/(x minus 3). Find the limit of (x squared minus 9) / (x minus 3) as x → 3.

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 3, so the limit equals its value there.

The answer. So the answer is Limit equals 6. 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(3.01) equals 6.01, f(3.001) equals 6.001, f(3.0001) equals 6.0001, f(3.00001) equals 6.00001, which checks out. Numerical values from the left agree: f(2.99) equals 5.99, f(2.999) equals 5.999, f(2.9999) equals 5.9999, f(2.99999) equals 5.99999, which checks out.

About limits

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

Find lim x->3 (x^2-9)/(x-3): Limits Step-by-Step Lesson