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Find lim x->infinity x^4e^(-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
$\lim_{x \to \infty} {x}^{4} {e}^{- x}$
2. Steps, part 1
1. $\lim_{x \to \infty} {x}^{4} {e}^{- x} = \lim_{x \to \infty} \frac{{x}^{4}}{{e}^{x}}$
  Indeterminate form 0·∞ — rewrite as a quotient
2. $\lim_{x \to \infty} \frac{{x}^{4}}{{e}^{x}} = \lim_{x \to \infty} \frac{4{x}^{3}}{{e}^{x}}$
  Indeterminate form ∞/∞ — apply L'Hôpital's rule
3. $\lim_{x \to \infty} \frac{4{x}^{3}}{{e}^{x}} = \lim_{x \to \infty} \frac{12{x}^{2}}{{e}^{x}}$
  Indeterminate form ∞/∞ — apply L'Hôpital's rule
4. $\lim_{x \to \infty} \frac{12{x}^{2}}{{e}^{x}} = \lim_{x \to \infty} \frac{24x}{{e}^{x}}$
  Indeterminate form ∞/∞ — apply L'Hôpital's rule
3. Steps, part 2
1. $\lim_{x \to \infty} \frac{24x}{{e}^{x}} = \lim_{x \to \infty} \frac{24x}{{e}^{x}}$
  Simplify
2. $\lim_{x \to \infty} \frac{24x}{{e}^{x}} = \lim_{x \to \infty} \frac{24}{{e}^{x}}$
  Indeterminate form ∞/∞ — apply L'Hôpital's rule
3. $\lim_{x \to \infty} \frac{24}{{e}^{x}} = \lim_{x \to \infty} \frac{24}{{e}^{x}}$
  Simplify
4. $\lim_{x \to \infty} \frac{24}{{e}^{x}} = 0$
  Bounded numerator over an unbounded denominator
4. The answer
$\lim_{x \to \infty} {x}^{4} {e}^{- x} = 0$
lim x→∞ x⁴ e^(-x) = 0
5. 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 to the power 4e to the power negative x. Find the limit of x to the power 4 e to the power negative x as x → ∞.

Steps, part 1. 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). Indeterminate form ∞/∞ — apply L prime Hôpital prime s rule. Differentiate the numerator and the denominator separately (this is not the quotient rule). Indeterminate form ∞/∞ — apply L prime Hôpital prime s rule. Differentiate the numerator and the denominator separately (this is not the quotient rule).

Steps, part 2. Simplify. 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 24 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.

Find lim x->infinity x^4e^(-x): Limits Step-by-Step Lesson