Question

determine the following limit
\\( \lim\limits_{x\to 2} \frac{x^3 - 7x^2 + 6x}{6 - x} \\)

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

\\( \bigcirc \\) a. \\( \lim\limits_{x\to 2} \frac{x^3 - 7x^2 + 6x}{6 - x} = \square \\) (simplify your answer.)
\\( \bigcirc \\) b. the limit does not exist.

Explanation:

Step1: Substitute x = 2 into the function

We substitute \( x = 2 \) into the numerator \( x^3 - 7x^2 + 6x \) and the denominator \( 6 - x \) separately.
For the numerator: \( 2^3 - 7\times2^2 + 6\times2 = 8 - 28 + 12 = -8 \)
For the denominator: \( 6 - 2 = 4 \)

Step2: Calculate the limit

Since the function is a rational function and the denominator is not zero at \( x = 2 \), we can directly substitute \( x = 2 \) into the function. So the limit is \( \frac{-8}{4} = -2 \)

Answer:

A. \( \lim\limits_{x\to 2}\frac{x^3 - 7x^2 + 6x}{6 - x} = -2 \)