Question

determine the following limit.
\\( \lim\limits_{x \to 5} \frac{25 - x^2}{x^2 - 7x + 10} \\)

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

\\( \bigcirc \\) a. \\( \lim\limits_{x \to 5} \frac{25 - x^2}{x^2 - 7x + 10} = \square \\) (simplify your answer.)
\\( \bigcirc \\) b. the limit does not exist.

Explanation:

Step1: Factor numerator and denominator

Numerator: \(25 - x^2=(5 - x)(5 + x)=-(x - 5)(x + 5)\)
Denominator: \(x^2-7x + 10=(x - 2)(x - 5)\)
So the function becomes \(\frac{-(x - 5)(x + 5)}{(x - 2)(x - 5)}\) (for \(x
eq5\)).

Step2: Cancel common factor

Cancel \((x - 5)\) (since \(x\to5\), \(x
eq5\) at limit point), get \(\frac{-(x + 5)}{x - 2}\).

Step3: Substitute \(x = 5\)

Substitute \(x = 5\) into \(\frac{-(x + 5)}{x - 2}\): \(\frac{-(5 + 5)}{5 - 2}=\frac{-10}{3}\).

Answer:

A. \(\lim\limits_{x\to5}\frac{25 - x^2}{x^2 - 7x + 10}=-\frac{10}{3}\)