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LimitsCalculusCollegeabout 1:38

Find lim x->infinity (3x^2+1)/(x^2-2): animated lesson

A narrated, step-by-step limits video lesson with captions. Final answer: Limit = 3.

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→∞3x2+1x2−2\lim_{x \to \infty} \frac{3{x}^{2}+1}{{x}^{2}-2}
  2. 2. Step by step
    1. lim⁡x→∞3x2+1x2−2=lim⁡x→∞6x2x\lim_{x \to \infty} \frac{3{x}^{2}+1}{{x}^{2}-2} = \lim_{x \to \infty} \frac{6x}{2x}
      Indeterminate form ∞/∞ — apply L'Hôpital's rule
    2. lim⁡x→∞6x2x=3\lim_{x \to \infty} \frac{6x}{2x} = 3
      Simplify
    3. lim⁡x→∞3=3\lim_{x \to \infty} 3 = 3
      The expression is constant
  3. 3. The answer
    lim⁡x→∞3x2+1x2−2=3\lim_{x \to \infty} \frac{3{x}^{2}+1}{{x}^{2}-2} = 3

    lim x→∞ (3 x² + 1) / (x² - 2) = 3

  4. 4. Check
    • Numerical values as x grows agree: f(127.18281) = 3.0004328, f(1271.8281) = 3.0000043, f(12718.281) = 3, f(127182.81) = 3 ✓

Transcript

The problem. Here is the problem: (3x squared plus 1)/(x squared minus 2). Find the limit of (3 x squared plus 1) / (x squared minus 2) 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. The expression is constant.

The answer. So the answer is Limit equals 3. 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 3.0004328, f(1271.8281) equals 3.0000043, f(12718.281) equals 3, f(127182.81) equals 3, 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 (3x^2+1)/(x^2-2): Limits Lesson