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Find lim x->1 ln(x)/(x-1): animated lesson

A narrated, step-by-step limits video lesson with captions. Final answer: Limit = 1.

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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 (0.8 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
$\lim_{x \to 1} \frac{\ln\left( x\right)}{x-1}$
2. Step by step
1. $\lim_{x \to 1} \frac{\ln\left( x\right)}{x-1} = \lim_{x \to 1} \frac{\frac{1}{x}}{1}$
  Indeterminate form 0/0 — apply L'Hôpital's rule
2. $\lim_{x \to 1} \frac{\frac{1}{x}}{1} = \lim_{x \to 1} \frac{1}{x}$
  Simplify
3. $\lim_{x \to 1} \frac{1}{x} = \frac{1}{1} = 1$
  Direct substitution
3. The answer
$\lim_{x \to 1} \frac{\ln\left( x\right)}{x-1} = 1$
lim x→1 ln(x) / (x - 1) = 1
4. Check
- Numerical values from the right agree: f(1.01) = 0.99503309, f(1.001) = 0.99950033, f(1.0001) = 0.99995, f(1.00001) = 0.999995 ✓
- Numerical values from the left agree: f(0.99) = 1.0050336, f(0.999) = 1.0005003, f(0.9999) = 1.00005, f(0.99999) = 1.000005 ✓

Transcript

The problem. Here is the problem: ln(x)/(x minus 1). Find the limit of ln(x) / (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 1. 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 0.99503309, f(1.001) equals 0.99950033, f(1.0001) equals 0.99995, f(1.00001) equals 0.999995, which checks out. Numerical values from the left agree: f(0.99) equals 1.0050336, f(0.999) equals 1.0005003, f(0.9999) equals 1.00005, f(0.99999) equals 1.000005, which checks out.

About limits

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

Find lim x->1 ln(x)/(x-1): Limits Step-by-Step Lesson