Question

determine whether the following limit exists. compute the limit if it exists. lim x→2 -2x² + 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→2 -2x² + 4x / x - 2 = lim x→2 (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 the limit becomes $\lim_{x
ightarrow2}\frac{-2x(x - 2)}{x - 2}$.

Step2: Simplify the rational - expression

Since $x
eq2$ when taking the limit (we are considering values of $x$ approaching 2, not equal to 2), we can cancel out the common factor $(x - 2)$ in the numerator and the denominator. The simplified expression is $\lim_{x
ightarrow2}-2x$.

Step3: Evaluate the limit

Substitute $x = 2$ into $-2x$. We have $-2\times2=-4$.

Answer:

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