L'Hopital's Rule Math Example 3

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Example 3

easy
Find lim⁡x→1x2−1x−1\displaystyle\lim_{x \to 1} \frac{x^2 - 1}{x - 1} using L'Hôpital's rule.

Solution

  1. 1
    1−11−1=00\frac{1-1}{1-1} = \frac{0}{0}, so apply L'Hôpital.
  2. 2
    lim⁡x→12x1=2\lim_{x\to1}\frac{2x}{1} = 2.

Answer

22
L'Hôpital applies since the form is 0/00/0. Differentiate numerator and denominator separately. (Factoring would also work: (x−1)(x+1)x−1→x+1→2\frac{(x-1)(x+1)}{x-1} \to x+1 \to 2.)

About L'Hopital's Rule

If lim⁡x→af(x)g(x)\lim_{x \to a} \frac{f(x)}{g(x)} is an indeterminate form 00\frac{0}{0} or ∞∞\frac{\infty}{\infty}, then lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x)\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} provided the right-hand limit exists (or is ±∞\pm\infty).

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More L'Hopital's Rule Examples

Example 1 easy

Find [formula] using L'Hôpital's rule.

Example 2 hard

Find [formula].

Example 4 medium

Find [formula].

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