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Chapter 2: Problem 21

Use the precise definition of a limit to prove the following limits. $$\lim _{x \rightarrow 4} \frac{x^{2}-16}{x-4}=8 \text { (Hint: Factor and simplify.) }$$

Short Answer

Expert verified
Question: Using the precise definition of a limit, prove that $\lim_{x \rightarrow 4}\frac{x^{2}-16}{x-4} = 8$. Answer: The limit is 8, as proved through factoring and simplifying the expression, then replacing x with 4 in the simplified expression.

Step by step solution

01

Factor the expression

Begin by factoring the numerator of the given expression. We have: $$\frac{x^{2}-16}{x-4} = \frac{(x+4)(x-4)}{x-4}$$
02

Simplify the expression

Now, eliminate the common factors in both the numerator and denominator: $$\frac{(x+4)(x-4)}{x-4} = x+4$$
03

Apply the limit definition

With the simplified expression, apply the limit: $$\lim_{x \rightarrow 4} (x+4)$$
04

Substitute the limiting value

Replace x with 4 in the expression: $$\lim _{x \rightarrow 4} (x+4) = 4 + 4$$
05

Calculate the limit

Finally, evaluate the limit: $$4+4 = 8$$ The proof is complete: the limit is indeed 8, as the exercise stated.

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