Question

find the limit.

$$\lim_{x \to 39} \frac{x - 39}{\sqrt{x + 10} - 7}$$

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. $$\lim_{x \to 39} \frac{x - 39}{\sqrt{x + 10} - 7} = \square$$ (type an integer or a simplified fraction.)

b. the limit does not exist.

Explanation:

Step1: Rationalize the denominator

Multiply the numerator and denominator by the conjugate of the denominator \(\sqrt{x + 10}+7\).

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Using the difference - of - squares formula \((a - b)(a + b)=a^{2}-b^{2}\), where \(a=\sqrt{x + 10}\) and \(b = 7\), the denominator becomes \((\sqrt{x+10})^{2}-7^{2}=x + 10-49=x - 39\).
So the limit becomes \(\lim_{x
ightarrow39}\frac{(x - 39)(\sqrt{x + 10}+7)}{x - 39}\).

Step2: Simplify the expression

Cancel out the common factor \(x - 39\) (since \(x
eq39\) when taking the limit as \(x
ightarrow39\)).
We get \(\lim_{x
ightarrow39}(\sqrt{x + 10}+7)\).

Step3: Substitute \(x = 39\)

Substitute \(x = 39\) into \(\sqrt{x + 10}+7\).
\(\sqrt{39+10}+7=\sqrt{49}+7=7 + 7=14\).

Answer:

\(14\)