Question

In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.1. lim (x → -2) (x + 2) / (x² - 4)

Accepted Answer

First, identify the form of the limit by substituting $$x = -2$$ into the expression $$\frac{x + 2}{x$$^{2}$$ - 4}. $$Check if it results in an indeterminate form like $$\frac{0}{0} or$$ $$\frac{\infty}{\infty}$$, which allows the use of l'Hôpital's Rule. Since direct substitution gives $$\frac{0}{0}$$, apply l'Hôpital's Rule by differentiating the numerator and denominator separately with respect to $$x. $$The derivative of the numerator $$x + 2 is$$ $$1$$, and the derivative of the denominator $$x^{2} - 4$ is 2x$. Rewrite the limit using these derivatives: $$\lim_{x 	o -2} \frac{1}{2x}$$. Now, substitute x = -2$ into this new expression to find the limit. To verify the result using a method from Chapter 2, factor the denominator x$$^{2}$$ - 4 as$$ $$(x - 2)(x + 2). $$Then simplify the original expression $$\frac{x + 2}{(x - 2)(x + 2)} by $$canceling the common factor $$(x + 2)$$, keeping in mind the domain restrictions. After simplification, evaluate the limit of the simplified expression as $$x$$ approaches $$-2 by $$direct substitution, confirming the result obtained using l'Hôpital's Rule.

In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.
1. lim (x → -2) (x + 2) / (x² - 4)

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Concetti chiave

Limits and Indeterminate Forms

l’Hôpital’s Rule

Algebraic Simplification of Limits