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LimitsCalculusCollegeabout 2:01

Find lim x->2 (x^3 - 8)/(x - 2): animated lesson

A narrated, step-by-step limits video lesson with captions. Final answer: Limit = 12.

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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
$\lim_{x \to 2} \frac{{x}^{3}-8}{x-2}$
2. Step by step
1. $\lim_{x \to 2} \frac{{x}^{3}-8}{x-2} = \lim_{x \to 2} \frac{3{x}^{2}}{1}$
  Indeterminate form 0/0 — apply L'Hôpital's rule
2. $\lim_{x \to 2} \frac{3{x}^{2}}{1} = \lim_{x \to 2} 3{x}^{2}$
  Simplify
3. $\lim_{x \to 2} 3{x}^{2} = 3\cdot{2}^{2} = 12$
  Direct substitution
3. The answer
$\lim_{x \to 2} \frac{{x}^{3}-8}{x-2} = 12$
lim x→2 (x³ - 8) / (x - 2) = 12
4. Check
- Numerical values from the right agree: f(2.01) = 12.0601, f(2.001) = 12.006001, f(2.0001) = 12.0006, f(2.00001) = 12.00006 ✓
- Numerical values from the left agree: f(1.99) = 11.9401, f(1.999) = 11.994001, f(1.9999) = 11.9994, f(1.99999) = 11.99994 ✓

Transcript

The problem. Here is the problem: (x cubed minus 8)/(x minus 2). Find the limit of (x cubed minus 8) / (x minus 2) as x → 2.

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 2, so the limit equals its value there.

The answer. So the answer is Limit equals 12. 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(2.01) equals 12.0601, f(2.001) equals 12.006001, f(2.0001) equals 12.0006, f(2.00001) equals 12.00006, which checks out. Numerical values from the left agree: f(1.99) equals 11.9401, f(1.999) equals 11.994001, f(1.9999) equals 11.9994, f(1.99999) equals 11.99994, which checks out.

About limits

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

Find lim x->2 (x^3 - 8)/(x - 2): Limits Step-by-Step Lesson