Question

determine whether the following limit exists. compute the limit if it exists.
lim_{x
ightarrow2}\frac{-2x^{2}+4x}{x - 2}
if possible, rewrite the limit by simplifying the rational expression. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. (lim_{x
ightarrow2}\frac{-2x^{2}+4x}{x - 2}=lim_{x
ightarrow2}square) (simplify your answer.)
b. the limit cannot be rewritten.

Explanation:

Step1: Factor the numerator

Factor out - 2x from the numerator $-2x^{2}+4x$. We get $-2x(x - 2)$. So, $\lim_{x
ightarrow2}\frac{-2x^{2}+4x}{x - 2}=\lim_{x
ightarrow2}\frac{-2x(x - 2)}{x - 2}$.

Step2: Simplify the rational - expression

Cancel out the common factor $(x - 2)$ in the numerator and the denominator (since $x
eq2$ when taking the limit). We have $\lim_{x
ightarrow2}\frac{-2x(x - 2)}{x - 2}=\lim_{x
ightarrow2}-2x$.

Step3: Evaluate the limit

Substitute $x = 2$ into $-2x$. $\lim_{x
ightarrow2}-2x=-2\times2=-4$.

Answer:

A. $\lim_{x
ightarrow2}\frac{-2x^{2}+4x}{x - 2}=\lim_{x
ightarrow2}-2x=-4$