Use l’Hôpital’s rule to find the limits in Exercises 7–52.


7. lim (x → 2) (x - 2) / (x² - 4)

Verified step by step guidance

1. First, recognize that the limit is of the form \( \frac{0}{0} \) when \( x \to 2 \), since substituting \( x = 2 \) gives \( \frac{2 - 2}{2^2 - 4} = \frac{0}{0} \). This indeterminate form allows us to apply l'Hôpital's Rule.
2. Recall l'Hôpital's Rule states that if \( \lim_{x \to a} \frac{f(x)}{g(x)} \) results in \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), then \( \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \), provided the latter limit exists.
3. Differentiate the numerator and denominator separately: \( f(x) = x - 2 \) so \( f'(x) = 1 \), and \( g(x) = x^2 - 4 \) so \( g'(x) = 2x \).
4. Rewrite the limit using the derivatives: \( \lim_{x \to 2} \frac{f'(x)}{g'(x)} = \lim_{x \to 2} \frac{1}{2x} \).
5. Finally, evaluate the new limit by substituting \( x = 2 \) into the expression \( \frac{1}{2x} \), which will give the value of the original limit.

Key Concepts

Limits and Indeterminate Forms

Limits describe the behavior of a function as the input approaches a certain value. When direct substitution results in an indeterminate form like 0/0, special techniques such as l’Hôpital’s rule are needed to evaluate the limit.

l’Hôpital’s Rule

l’Hôpital’s rule states that if a limit yields an indeterminate form 0/0 or ∞/∞, the limit of the ratio of the functions equals the limit of the ratio of their derivatives, provided this latter limit exists.

Derivative of Functions

Derivatives measure the instantaneous rate of change of a function. Applying l’Hôpital’s rule requires finding the derivatives of the numerator and denominator functions to simplify the limit evaluation.