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LimitsCalculusCollegeabout 1:49

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

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

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 (0.6 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    lim⁡x→∞x2+1x+1\lim_{x \to \infty} \frac{{x}^{2}+1}{x+1}
  2. 2. Step by step
    1. lim⁡x→∞x2+1x+1=lim⁡x→∞2x1\lim_{x \to \infty} \frac{{x}^{2}+1}{x+1} = \lim_{x \to \infty} \frac{2x}{1}
      Indeterminate form ∞/∞ — apply L'Hôpital's rule
    2. lim⁡x→∞2x1=lim⁡x→∞2x\lim_{x \to \infty} \frac{2x}{1} = \lim_{x \to \infty} 2x
      Simplify
    3. lim⁡x→∞2x=∞\lim_{x \to \infty} 2x = \infty
      The expression grows without bound
    4. Conclusion
  3. 3. The answer
    lim⁡x→∞x2+1x+1=∞\lim_{x \to \infty} \frac{{x}^{2}+1}{x+1} = \infty

    lim x→∞ (x² + 1) / (x + 1) = ∞

  4. 4. Check
    • Numerical values as x grows agree: f(127.18281) = 126.19841, f(1271.8281) = 1270.8297, f(12718.281) = 12717.281, f(127182.81) = 127181.81 ✓

Transcript

The problem. Here is the problem: (x squared plus 1)/(x plus 1). Find the limit of (x squared plus 1) / (x plus 1) 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 grows without bound. As x → ∞, the values increase in size without limit. Conclusion. The function is unbounded, so the limit is +∞ (strictly, the limit does not exist as a real number; it diverges to +∞).

The answer. So the answer is Limit equals ∞. 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 126.19841, f(1271.8281) equals 1270.8297, f(12718.281) equals 12717.281, f(127182.81) equals 127181.81, 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^2 + 1)/(x + 1): Limits Lesson