find the limit if it exists.\n\$\\lim_{x\\to8}x(x - 4)\$\n\na. \$(\\lim_{x\\to8}x)\\cdot(x - 4)\$\nb…

find the limit if it exists.\n\$\\lim_{x\\to8}x(x - 4)\$\n\na. \$(\\lim_{x\\to8}x)\\cdot(x - 4)\$\nb. \$\\lim_{x\\to8}x\\cdot(\\lim_{x\\to8}x-\\lim_{x\\to8}4)\$\nc. \$\\lim_{x\\to8}x+(\\lim_{x\\to8}x-\\lim_{x\\to8}4)\$\nd. \$x\\cdot(\\lim_{x\\to8}x-\\lim_{x\\to8}4)\$\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. \$\\lim_{x\\to8}x(x - 4)=\$ (simplify your answer.)\nb. the limit does not exist.

Answer

Explanation:

Step1: Apply limit - product rule

The limit - product rule states that $\lim_{x\rightarrow a}(f(x)\cdot g(x))=\lim_{x\rightarrow a}f(x)\cdot\lim_{x\rightarrow a}g(x)$. Here $f(x)=x$ and $g(x)=x - 4$. So, $\lim_{x\rightarrow8}x(x - 4)=\lim_{x\rightarrow8}x\cdot\lim_{x\rightarrow8}(x - 4)$. And by the limit - difference rule $\lim_{x\rightarrow a}(f(x)-g(x))=\lim_{x\rightarrow a}f(x)-\lim_{x\rightarrow a}g(x)$, we have $\lim_{x\rightarrow8}(x - 4)=\lim_{x\rightarrow8}x-\lim_{x\rightarrow8}4$. So $\lim_{x\rightarrow8}x(x - 4)=\lim_{x\rightarrow8}x\cdot(\lim_{x\rightarrow8}x-\lim_{x\rightarrow8}4)$.

Step2: Evaluate individual limits

We know that $\lim_{x\rightarrow8}x = 8$ and $\lim_{x\rightarrow8}4=4$. Then $\lim_{x\rightarrow8}x\cdot(\lim_{x\rightarrow8}x-\lim_{x\rightarrow8}4)=8\cdot(8 - 4)$.

Step3: Calculate the result

$8\cdot(8 - 4)=8\times4 = 32$.

Answer:

B. $\lim_{x\rightarrow8}x\cdot(\lim_{x\rightarrow8}x-\lim_{x\rightarrow8}4)$ A. $\lim_{x\rightarrow8}x(x - 4)=32$