QuestionFind the limit: $\lim _{u \rightarrow-5} \frac{u+5}{u^{3}+125}$ and simplify the expression.

Studdy Solution

STEP 1

Assumptions1. We are given the limit as $u$ approaches -5 of the function $\frac{u+5}{u^{3}+125}$ . . We need to simplify the rational expression as much as possible.

STEP 2

First, we need to factor the denominator $u^{}+125$ .
The sum of cubes formula is $a^ + b^ = (a+b)(a^2 - ab + b^2)$ . Here, $a=u$ and $b=5$ .
So, we can write $u^{}+125$ as $(u+5)(u^2 -5u +25)$ .

STEP 3

Substitute the factored form of the denominator into the limit expression.
$\lim{u \rightarrow-5} \frac{u+5}{(u+5)(u^2 -5u +25)}$

STEP 4

We can now cancel out the common factor of $u+$ in the numerator and the denominator.
$\lim{u \rightarrow-} \frac{1}{u^2 -u +25}$

STEP 5

Now, we can substitute $u=-5$ into the simplified limit expression.
$\lim{u \rightarrow-5} \frac{1}{(-5)^2 -5(-5) +25}$

STEP 6

Calculate the value of the limit.
$\lim{u \rightarrow-5} \frac{1}{25 +25 +25} = \frac{1}{75}$ The simplified rational expression as $u$ approaches -5 is $\frac{1}{75}$ .

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