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} (\square) \\) (simplify your answer.)

Explanation:

Step1: Factor numerator and denominator

Factor \(w^2 + 6kw + 5k^2\) as \((w + k)(w + 5k)\) and \(w^2 + kw\) as \(w(w + k)\).
So the expression becomes \(\frac{(w + k)(w + 5k)}{w(w + k)}\).

Step2: Cancel common factor

Cancel the common factor \((w + k)\) (since \(w\to -k\) but \(w
eq -k\) when taking the limit, so cancellation is valid).
We get \(\frac{w + 5k}{w}\).

Answer:

\(\frac{w + 5k}{w}\)