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LimitsCalculusCollegeabout 2:13

Find lim x->0 x^2 sin(1/x): animated lesson

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

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. 1. The problem
    Worked example
    lim⁡x→0x2sin⁡(1x)\lim_{x \to 0} {x}^{2} \sin\left(\frac{1}{x}\right)
  2. 2. Step by step
    1. lim⁡x→0x2sin⁡(1x)=0\lim_{x \to 0} {x}^{2} \sin\left(\frac{1}{x}\right) = 0
      Evaluate the limiting behaviour
    2. lim⁡x→0x2sin⁡(1x)=0\lim_{x \to 0} {x}^{2} \sin\left(\frac{1}{x}\right) = 0
      Identify the value
  3. 3. The answer
    lim⁡x→0x2sin⁡(1x)=0\lim_{x \to 0} {x}^{2} \sin\left(\frac{1}{x}\right) = 0

    lim x→0 x² sin(1 / x) = 0

  4. 4. Check
    • Numerical values from the right agree: f(0.01) = -0.000050636564, f(0.001) = 8.268795e-7, f(0.0001) = -3.056144e-9, f(0.00001) = 3.57488e-12 ✓
    • Numerical values from the left agree: f(-0.01) = 0.000050636564, f(-0.001) = -8.268795e-7, f(-0.0001) = 3.056144e-9, f(-0.00001) = -3.57488e-12 ✓

Transcript

The problem. Here is the problem: x squared sin(1/x). Find the limit of x squared sin(1 / x) as x → 0.

Step by step. Evaluate the limiting behaviour. The dominant behaviour of each piece determines the limit. Identify the value. The rewritten expression settles to 0, which equals 0 to within 10 to the power minus 6; the converging numerical values below confirm it.

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 agree: f(0.01) equals negative 0.000050636564, f(0.001) equals 8.268795e minus 7, f(0.0001) equals negative 3.056144e minus 9, f(0.00001) equals 3.57488e minus 12, which checks out. Numerical values from the left agree: f(negative 0.01) equals 0.000050636564, f(negative 0.001) equals negative 8.268795e minus 7, f(negative 0.0001) equals 3.056144e minus 9, f(negative 0.00001) equals negative 3.57488e minus 12, which checks out.

About limits

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

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Find lim x->0 x^2 sin(1/x): Limits Step-by-Step Lesson