Question

find the following limit or state that it does not exist. assume k is a fixed real number.

$$\\lim_{w \\to -k} \\frac{w^2 + 6kw + 5k^2}{w^2 + kw}, k \ eq 0$$

simplify the given limit.

$$\\lim_{w \\to -k} \\frac{w^2 + 6kw + 5k^2}{w^2 + kw} = \\lim_{w \\to -k} \\left( \\frac{w + 5k}{w} \ ight) \\text{ (simplify your answer.)}$$

evaluate the limit, if possible. select the correct choice and, if necessary, fill in the answer box to complete your choice.
a. $$\lim_{w \to -k} \frac{w^2 + 6kw + 5k^2}{w^2 + kw} = \square \text{ (type an exact answer.)}$$
b. the limit does not exist.

Explanation:

Step1: Factor numerator and denominator

Numerator: \(w^2 + 6kw + 5k^2=(w + k)(w + 5k)\)
Denominator: \(w^2 + kw = w(w + k)\)
So the limit becomes \(\lim_{w\to -k}\frac{(w + k)(w + 5k)}{w(w + k)}\) (cancel \(w + k\) as \(w
eq -k\) when approaching)

Step2: Simplify and substitute \(w=-k\)

Simplified: \(\lim_{w\to -k}\frac{w + 5k}{w}\)
Substitute \(w = -k\): \(\frac{-k + 5k}{-k}=\frac{4k}{-k}=-4\)

Answer:

A. \(\lim\limits_{w\to -k}\frac{w^2 + 6kw + 5k^2}{w^2 + kw}=-4\)