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LimitsCalculusCollegeabout 2:07

Find lim x->infinity e^(-x) x^3: 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 (0.7 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
$\lim_{x \to \infty} {e}^{- x} {x}^{3}$
2. Steps, part 1
1. $\lim_{x \to \infty} {e}^{- x} {x}^{3} = \lim_{x \to \infty} \frac{{x}^{3}}{{e}^{x}}$
  Indeterminate form 0·∞ — rewrite as a quotient
2. $\lim_{x \to \infty} \frac{{x}^{3}}{{e}^{x}} = \lim_{x \to \infty} \frac{3{x}^{2}}{{e}^{x}}$
  Indeterminate form ∞/∞ — apply L'Hôpital's rule
3. $\lim_{x \to \infty} \frac{3{x}^{2}}{{e}^{x}} = \lim_{x \to \infty} \frac{6x}{{e}^{x}}$
  Indeterminate form ∞/∞ — apply L'Hôpital's rule
4. $\lim_{x \to \infty} \frac{6x}{{e}^{x}} = \lim_{x \to \infty} \frac{6x}{{e}^{x}}$
  Simplify
3. Steps, part 2
1. $\lim_{x \to \infty} \frac{6x}{{e}^{x}} = \lim_{x \to \infty} \frac{6}{{e}^{x}}$
  Indeterminate form ∞/∞ — apply L'Hôpital's rule
2. $\lim_{x \to \infty} \frac{6}{{e}^{x}} = \lim_{x \to \infty} \frac{6}{{e}^{x}}$
  Simplify
3. $\lim_{x \to \infty} \frac{6}{{e}^{x}} = 0$
  Bounded numerator over an unbounded denominator
4. The answer
$\lim_{x \to \infty} {e}^{- x} {x}^{3} = 0$
lim x→∞ e^(-x) x³ = 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: e to the power negative x x cubed. Find the limit of e to the power negative x x cubed 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). Simplify.

Steps, part 2. 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 6 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 e^(-x) x^3: Limits Step-by-Step Lesson