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LimitsCalculusCollegeabout 2:13

Find lim x->0 x ln(x): animated lesson

A narrated, step-by-step limits video lesson with captions. Final answer: Limit = 0.

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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 (0.9 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
$\lim_{x \to 0} x \ln\left( x\right)$
2. Step by step
1. $\lim_{x \to 0} x \ln\left( x\right) = \lim_{x \to 0} \frac{\ln\left( x\right)}{\frac{1}{x}}$
  Indeterminate form 0·∞ — rewrite as a quotient
2. $\lim_{x \to 0} \frac{\ln\left( x\right)}{\frac{1}{x}} = \lim_{x \to 0} \frac{\frac{1}{x}}{-\frac{1}{{x}^{2}}}$
  Indeterminate form ∞/∞ — apply L'Hôpital's rule
3. $\lim_{x \to 0} \frac{\frac{1}{x}}{-\frac{1}{{x}^{2}}} = \lim_{x \to 0} - x$
  Simplify
4. $\lim_{x \to 0} - x = -0 = 0$
  Direct substitution
3. The answer
$\lim_{x \to 0} x \ln\left( x\right) = 0$
lim x→0 x ln(x) = 0
4. Check
- Numerical values from the right: f(0.01) = -0.046051702, f(0.001) = -0.0069077553, f(0.0001) = -0.00092103404, f(0.00001) = -0.00011512925 (no clear trend at this precision)
- Numerical values from the left: f(-0.01) = undefined, f(-0.001) = undefined, f(-0.0001) = undefined, f(-0.00001) = undefined (no clear trend at this precision)

Transcript

The problem. Here is the problem: x ln(x). Find the limit of x ln(x) as x → 0.

Step by step. Indeterminate form 0 times ∞ — rewrite as a quotient. Moving one factor to the denominator as its reciprocal turns 0 times ∞ into 0 over 0 or ∞/∞. Indeterminate form ∞/∞ — 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 0, so the limit equals its value there.

The answer. So the answer is Limit equals 0. 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: f(0.01) equals negative 0.046051702, f(0.001) equals negative 0.0069077553, f(0.0001) equals negative 0.00092103404, f(0.00001) equals negative 0.00011512925 (no clear trend at this precision). Numerical values from the left: f(negative 0.01) equals undefined, f(negative 0.001) equals undefined, f(negative 0.0001) equals undefined, f(negative 0.00001) equals undefined (no clear trend at this precision).

About limits

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

Find lim x->0 x ln(x): Animated Limits Lesson, Step by Step