Question

find the limit if it exists.
\\(\lim_{x\to8}x(x - 4)\\)

a. \\((\lim_{x\to8}x)\cdot(x - 4)\\)
b. \\(\lim_{x\to8}x\cdot(\lim_{x\to8}x-\lim_{x\to8}4)\\)
c. \\(\lim_{x\to8}x+(\lim_{x\to8}x-\lim_{x\to8}4)\\)
d. \\(x\cdot(\lim_{x\to8}x-\lim_{x\to8}4)\\)

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

a. \\(\lim_{x\to8}x(x - 4)=\\) (simplify your answer.)
b. the limit does not exist.

Explanation:

Step1: Apply limit - product rule

The limit - product rule states that $\lim_{x
ightarrow a}(f(x)\cdot g(x))=\lim_{x
ightarrow a}f(x)\cdot\lim_{x
ightarrow a}g(x)$. Here $f(x)=x$ and $g(x)=x - 4$. So, $\lim_{x
ightarrow8}x(x - 4)=\lim_{x
ightarrow8}x\cdot\lim_{x
ightarrow8}(x - 4)$. And by the limit - difference rule $\lim_{x
ightarrow a}(f(x)-g(x))=\lim_{x
ightarrow a}f(x)-\lim_{x
ightarrow a}g(x)$, we have $\lim_{x
ightarrow8}(x - 4)=\lim_{x
ightarrow8}x-\lim_{x
ightarrow8}4$. So $\lim_{x
ightarrow8}x(x - 4)=\lim_{x
ightarrow8}x\cdot(\lim_{x
ightarrow8}x-\lim_{x
ightarrow8}4)$.

Step2: Evaluate individual limits

We know that $\lim_{x
ightarrow8}x = 8$ and $\lim_{x
ightarrow8}4=4$. Then $\lim_{x
ightarrow8}x\cdot(\lim_{x
ightarrow8}x-\lim_{x
ightarrow8}4)=8\cdot(8 - 4)$.

Step3: Calculate the result

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

Answer:

B. $\lim_{x
ightarrow8}x\cdot(\lim_{x
ightarrow8}x-\lim_{x
ightarrow8}4)$
A. $\lim_{x
ightarrow8}x(x - 4)=32$