9. find the limit.
$\lim_{x\to\infty}(\sqrt{x + 9}-\sqrt{x + 4})$

Question

Accepted Answer

# Explanation:
## Step1: Rationalize the expression
Multiply and divide by the conjugate $\sqrt{x + 9}+\sqrt{x + 4}$.
$$
\begin{align*}
&\lim_{x
ightarrow\infty}(\sqrt{x + 9}-\sqrt{x + 4})\times\frac{\sqrt{x + 9}+\sqrt{x + 4}}{\sqrt{x + 9}+\sqrt{x + 4}}\
=&\lim_{x
ightarrow\infty}\frac{(\sqrt{x + 9})^2-(\sqrt{x + 4})^2}{\sqrt{x + 9}+\sqrt{x + 4}}\
=&\lim_{x
ightarrow\infty}\frac{(x + 9)-(x + 4)}{\sqrt{x + 9}+\sqrt{x + 4}}\
=&\lim_{x
ightarrow\infty}\frac{5}{\sqrt{x + 9}+\sqrt{x + 4}}
\end{align*}
$$
## Step2: Analyze the limit as $x
ightarrow\infty$
As $x
ightarrow\infty$, both $\sqrt{x + 9}
ightarrow\infty$ and $\sqrt{x + 4}
ightarrow\infty$.
So, $\lim_{x
ightarrow\infty}\frac{5}{\sqrt{x + 9}+\sqrt{x + 4}} = 0$.

# Answer:
$0$

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Question

Explanation:

Step1: Rationalize the expression

Step2: Analyze the limit as $x

Answer: