Questions: Find the following limit or state that it does not exist. Assume (k) is a fixed nonzero real number. [ lim w rightarrow-k fracw^2+5 k w+4 k^2w^2+k w ] Evaluate the limit, if possible. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. (lim w rightarrow-k fracw^2+5 k w+4 k^2w^2+k w=) (Type an exact answer.) B. The limit does not exist.

$Find the following limit or state that it does not exist. Assume (k) is a fixed nonzero real number. [ lim w rightarrow-k fracw^2+5 k w+4 k^2w^2+k w ] Evaluate the limit, if possible. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. (lim w rightarrow-k fracw^2+5 k w+4 k^2w^2+k w=) (Type an exact answer.) B. The limit does not exist.$

Transcript text: Find the following limit or state that it does not exist. Assume $k$ is a fixed nonzero real number. \[ \lim _{w \rightarrow-k} \frac{w^{2}+5 k w+4 k^{2}}{w^{2}+k w} \] Evaluate the limit, if possible. Select the correct choice below and, if necessary, fill in the answer box to com choice. A. $\lim _{w \rightarrow-k} \frac{w^{2}+5 k w+4 k^{2}}{w^{2}+k w}=\square$ (Type an exact answer.) B. The limit does not exist.

Solution

Solution Steps

To evaluate the limit, we first check if direct substitution of $ w = -k $ leads to an indeterminate form. If it does, we can try to simplify the expression by factoring or other algebraic techniques. If simplification resolves the indeterminate form, we can then substitute $ w = -k $ again to find the limit.

Step 1: Check for Indeterminate Form

First, we substitute $ w = -k $ into the expression: \[ \frac{(-k)^2 + 5k(-k) + 4k^2}{(-k)^2 + k(-k)} = \frac{k^2 - 5k^2 + 4k^2}{k^2 - k^2} = \frac{0}{0} \] This results in an indeterminate form $ \frac{0}{0} $.

Step 2: Simplify the Expression

To resolve the indeterminate form, we factor both the numerator and the denominator: \[ \frac{w^2 + 5kw + 4k^2}{w^2 + kw} = \frac{(w + 4k)(w + k)}{w(w + k)} \] We can cancel the common factor $ (w + k) $ in the numerator and the denominator: \[ \frac{(w + 4k)(w + k)}{w(w + k)} = \frac{w + 4k}{w} \]

Step 3: Evaluate the Limit

Now, we substitute $ w = -k $ into the simplified expression: \[ \lim_{w \to -k} \frac{w + 4k}{w} = \frac{-k + 4k}{-k} = \frac{3k}{-k} = -3 \]