find the limit.\n\n$$\\lim_{x \\to 39} \\frac{x - 39}{\\sqrt{x + 10} - 7}$$\n\nselect the correct choice…

find the limit.\n\n$$\\lim_{x \\to 39} \\frac{x - 39}{\\sqrt{x + 10} - 7}$$\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. $$\\lim_{x \\to 39} \\frac{x - 39}{\\sqrt{x + 10} - 7} = \\square$$ (type an integer or a simplified fraction.)\n\nb. the limit does not exist.

find the limit.\n\n$$\\lim_{x \\to 39} \\frac{x - 39}{\\sqrt{x + 10} - 7}$$\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. $$\\lim_{x \\to 39} \\frac{x - 39}{\\sqrt{x + 10} - 7} = \\square$$ (type an integer or a simplified fraction.)\n\nb. the limit does not exist.

Answer

Explanation:

Step1: Rationalize the denominator

Multiply the numerator and denominator by the conjugate of the denominator (\sqrt{x + 10}+7). [ \begin{align*} \lim_{x\rightarrow39}\frac{x - 39}{\sqrt{x + 10}-7}&=\lim_{x\rightarrow39}\frac{(x - 39)(\sqrt{x + 10}+7)}{(\sqrt{x + 10}-7)(\sqrt{x + 10}+7)}\ \end{align*} ] 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\rightarrow39}\frac{(x - 39)(\sqrt{x + 10}+7)}{x - 39}).

Step2: Simplify the expression

Cancel out the common factor (x - 39) (since (x\neq39) when taking the limit as (x\rightarrow39)). We get (\lim_{x\rightarrow39}(\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)