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Find lim x->infinity x sin(1/x): 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 (0.6 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
$\lim_{x \to \infty} x \sin\left(\frac{1}{x}\right)$
2. Step by step
1. $\lim_{x \to \infty} x \sin\left(\frac{1}{x}\right) = \lim_{x \to \infty} \frac{\sin\left(\frac{1}{x}\right)}{\frac{1}{x}}$
  Indeterminate form 0·∞ — rewrite as a quotient
2. $\lim_{x \to \infty} \frac{\sin\left(\frac{1}{x}\right)}{\frac{1}{x}} = \lim_{x \to \infty} \frac{-\frac{\cos\left(\frac{1}{x}\right)}{{x}^{2}}}{-\frac{1}{{x}^{2}}}$
  Indeterminate form 0/0 — apply L'Hôpital's rule
3. $\lim_{x \to \infty} \frac{-\frac{\cos\left(\frac{1}{x}\right)}{{x}^{2}}}{-\frac{1}{{x}^{2}}} = \lim_{x \to \infty} \cos\left(\frac{1}{x}\right)$
  Simplify
4. $\lim_{x \to \infty} \cos\left(\frac{1}{x}\right) = \lim_{t \to 0} \cos\left(\mathrm{t}\right) = 1$
  Substitute x = 1/t (t → 0⁺)
3. The answer
$\lim_{x \to \infty} x \sin\left(\frac{1}{x}\right) = 1$
lim x→∞ x sin(1 / x) = 1
4. Check
- Numerical values as x grows agree: f(127.18281) = 0.9999897, f(1271.8281) = 0.9999999, f(12718.281) = 1, f(127182.81) = 1 ✓

Transcript

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

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 0 over 0 — apply L prime Hôpital prime s rule. Differentiate the numerator and the denominator separately (this is not the quotient rule). Simplify. Substitute x equals 1/t (t → 0 to the power plus ).

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 as x grows agree: f(127.18281) equals 0.9999897, f(1271.8281) equals 0.9999999, f(12718.281) equals 1, f(127182.81) equals 1, which checks out.

About limits

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

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