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LimitsCalculusCollegeabout 2:00

Find lim x->infinity (2x+3)/(5x-1): animated lesson

A narrated, step-by-step limits video lesson with captions. Final answer: Limit = 2/5 ≈ 0.4.

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→∞2x+35x−1\lim_{x \to \infty} \frac{2x+3}{5x-1}
  2. 2. Step by step
    1. lim⁡x→∞2x+35x−1=lim⁡x→∞25\lim_{x \to \infty} \frac{2x+3}{5x-1} = \lim_{x \to \infty} \frac{2}{5}
      Indeterminate form ∞/∞ — apply L'Hôpital's rule
    2. lim⁡x→∞25=25\lim_{x \to \infty} \frac{2}{5} = \frac{2}{5}
      Simplify
    3. lim⁡x→∞25=25\lim_{x \to \infty} \frac{2}{5} = \frac{2}{5}
      The expression is constant
  3. 3. The answer
    lim⁡x→∞2x+35x−1=25≈0.4\lim_{x \to \infty} \frac{2x+3}{5x-1} = \frac{2}{5} \approx 0.4

    lim x→∞ (2 x + 3) / (5 x - 1) = 2/5 ≈ 0.4

  4. 4. Check
    • Numerical values as x grows agree: f(127.18281) = 0.40535506, f(1271.8281) = 0.40053475, f(12718.281) = 0.40005347, f(127182.81) = 0.40000535 ✓

Transcript

The problem. Here is the problem: (2x plus 3)/(5x minus 1). Find the limit of (2 x plus 3) / (5 x minus 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 is constant.

The answer. So the answer is Limit equals 2 over 5 approximately 0.4. 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 0.40535506, f(1271.8281) equals 0.40053475, f(12718.281) equals 0.40005347, f(127182.81) equals 0.40000535, 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 (2x+3)/(5x-1): Step-by-Step Lesson