Question

find the limit if it exists.
\\(\lim_{x\to8}x(x - 4)\\)
which of the following shows the correct expression after the limit properties have been applied?
o a. \\((\lim_{x\to8}x)\cdot(x - 4)\\)
o b. \\(\lim_{x\to8}x\cdot(\lim_{x\to8}x-\lim_{x\to8}4)\\)
o c. \\(\lim_{x\to8}x+(\lim_{x\to8}x-\lim_{x\to8}4)\\)
o d. \\(x\cdot(\lim_{x\to8}x-\lim_{x\to8}4)\\)

Explanation:

Step1: Recall product - rule of limits

The product - rule of limits 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 $g(x)=x - 4$. Also, the difference - rule of limits states that $\lim_{x
ightarrow a}(f(x)-g(x))=\lim_{x
ightarrow a}f(x)-\lim_{x
ightarrow a}g(x)$.
For $\lim_{x
ightarrow 8}x(x - 4)$, we first apply the product - rule: $\lim_{x
ightarrow 8}x(x - 4)=\lim_{x
ightarrow 8}x\cdot\lim_{x
ightarrow 8}(x - 4)$. Then we apply the difference - rule to $\lim_{x
ightarrow 8}(x - 4)$ which gives $\lim_{x
ightarrow 8}(x - 4)=\lim_{x
ightarrow 8}x-\lim_{x
ightarrow 8}4$. So $\lim_{x
ightarrow 8}x(x - 4)=\lim_{x
ightarrow 8}x\cdot(\lim_{x
ightarrow 8}x-\lim_{x
ightarrow 8}4)$.

Answer:

B. $\lim_{x
ightarrow 8}x\cdot(\lim_{x
ightarrow 8}x-\lim_{x
ightarrow 8}4)$