Question

homework2: problem 11 (1 point) results for this submission 2 of the answers are not correct. part 1: evaluate the limit evaluate the following limit by simplifying the expression (first answer box) and then evaluating the limit. $lim_{x \to 8}\frac{x - 8}{sqrt{x}-sqrt{8}}=lim_{x \to 8}(sqrt{x}+sqrt{8}) = 4sqrt{2}$ hint: treat $x - 8$ as a difference of squares. note: in your written solution, you should write the limit steps. part 2: follow - up question $lim_{x \to 8}\frac{sqrt{x}-sqrt{8}}{x - 8}=lim_{x \to 8}square=square$ note: you can earn partial credit on this problem. preview my answers submit answers your score was recorded. you have attempted this problem 2 times. you received a score of 50% for this attempt. your overall recorded score is 50%. you have unlimited attempts remaining.

Explanation:

Step1: Rationalize the numerator

Multiply the fraction $\frac{\sqrt{x}-\sqrt{8}}{x - 8}$ by $\frac{\sqrt{x}+\sqrt{8}}{\sqrt{x}+\sqrt{8}}$. We get $\lim_{x
ightarrow8}\frac{(\sqrt{x}-\sqrt{8})(\sqrt{x}+\sqrt{8})}{(x - 8)(\sqrt{x}+\sqrt{8})}$. Using the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$, the numerator becomes $x - 8$. So the expression is $\lim_{x
ightarrow8}\frac{x - 8}{(x - 8)(\sqrt{x}+\sqrt{8})}$.

Step2: Simplify the fraction

Cancel out the common factor $(x - 8)$ in the numerator and denominator (since $x
eq8$ when taking the limit), we have $\lim_{x
ightarrow8}\frac{1}{\sqrt{x}+\sqrt{8}}$.

Step3: Evaluate the limit

Substitute $x = 8$ into $\frac{1}{\sqrt{x}+\sqrt{8}}$. We get $\frac{1}{\sqrt{8}+\sqrt{8}}=\frac{1}{2\sqrt{8}}=\frac{1}{4\sqrt{2}}$.

Answer:

$\frac{1}{4\sqrt{2}}$