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LimitsCalculusCollegeabout 1:29

Find lim x->infinity x/e^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 (0.5 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    lim⁡x→∞xex\lim_{x \to \infty} \frac{x}{{e}^{x}}
  2. 2. Step by step
    1. lim⁡x→∞xex=lim⁡x→∞1ex\lim_{x \to \infty} \frac{x}{{e}^{x}} = \lim_{x \to \infty} \frac{1}{{e}^{x}}
      Indeterminate form ∞/∞ — apply L'Hôpital's rule
    2. lim⁡x→∞1ex=lim⁡x→∞1ex\lim_{x \to \infty} \frac{1}{{e}^{x}} = \lim_{x \to \infty} \frac{1}{{e}^{x}}
      Simplify
    3. lim⁡x→∞1ex=0\lim_{x \to \infty} \frac{1}{{e}^{x}} = 0
      Bounded numerator over an unbounded denominator
  3. 3. The answer
    lim⁡x→∞xex=0\lim_{x \to \infty} \frac{x}{{e}^{x}} = 0

    lim x→∞ x / e^x = 0

  4. 4. Check
    • Numerical values as x grows agree: f(127.18281) = 0, f(1271.8281) = 0, f(12718.281) = 0, f(127182.81) = 0 ✓

Transcript

The problem. Here is the problem: x/e to the power x. Find the limit of x / e to the power x as x → ∞.

Step by step. Indeterminate form ∞/∞ — apply L prime Hôpital prime s rule. Differentiate the numerator and the denominator separately (this is not the quotient rule). Simplify. Bounded numerator over an unbounded denominator. The numerator approaches 1 and the denominator grows without bound, so the quotient approaches 0.

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 as x grows agree: f(127.18281) equals 0, f(1271.8281) equals 0, f(12718.281) equals 0, f(127182.81) equals 0, which checks out.

About limits

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

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Find lim x->infinity x/e^x: Limits Step-by-Step Lesson