EZ Math

Calculus · Limits

How to Find a Limit at Infinity

Problem

lim⁡x→∞3x2+2xx2−1\lim_{x \to \infty} \frac{3x^2 + 2x}{x^2 - 1}

Answer

33

At infinity, only the fastest-growing terms matter. Dividing by the highest power shows which terms survive and which fade to zero.

Step-by-step solution

  1. Find the highest power of x in the denominator.

    x2x^2

    Why: Dividing by the highest power turns the large terms into constants.

  2. Divide every term by x squared.

    3+2x1−1x2\frac{3 + \frac{2}{x}}{1 - \frac{1}{x^2}}

    Why: Each division keeps the fraction equal to the original.

  3. Take the limit of each small fraction.

    2x→0,1x2→0\frac{2}{x} \to 0,\quad \frac{1}{x^2} \to 0

    Why: A fixed number divided by a growing number approaches zero.

  4. Evaluate what is left.

    3+01−0=3\frac{3 + 0}{1 - 0} = 3

    Why: The leading coefficients decide the limit.

Common mistakes

  • Cancelling x squared terms inside the sums without dividing every term.
  • Concluding the limit is infinity because both top and bottom grow. The powers match.
  • Forgetting the denominator constant 1. It stays.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

  1. lim⁡x→∞2x2+1x2+3\lim_{x \to \infty} \frac{2x^2 + 1}{x^2 + 3}

    Show answer

    22

  2. lim⁡x→∞5x3−x2x3+x2\lim_{x \to \infty} \frac{5x^3 - x}{2x^3 + x^2}

    Show answer

    52\frac{5}{2}

  3. lim⁡x→∞x+43x−1\lim_{x \to \infty} \frac{x + 4}{3x - 1}

    Show answer

    13\frac{1}{3}

  4. lim⁡x→∞4x2−2x2−7\lim_{x \to \infty} \frac{4x^2 - 2}{x^2 - 7}

    Show answer

    44

  5. lim⁡x→∞7x2x+5\lim_{x \to \infty} \frac{7x}{2x + 5}

    Show answer

    72\frac{7}{2}