Question

⑤ use properties of limits to evaluate the following limits if they exist
a $lim_{u
ightarrow - 2}sqrt{9 - u^{3}+2u^{3}}$

Explanation:

Step1: Substitute $u = - 2$ into the expression

First, consider the expression inside the square - root: $9 - u^{3}+2u^{3}=9 + u^{3}$.
When $u=-2$, we have $9+(-2)^{3}=9 - 8=1$.

Step2: Evaluate the square - root limit

The original limit is $\lim_{u
ightarrow - 2}\sqrt{9 - u^{3}+2u^{3}}=\lim_{u
ightarrow - 2}\sqrt{9 + u^{3}}$.
Since the function $y = \sqrt{9 + u^{3}}$ is continuous at $u=-2$, we can directly substitute $u = - 2$ into the function.
$\sqrt{9+(-2)^{3}}=\sqrt{1}=1$.

Answer:

$1$