Question

1. find the limit.

\\(\lim_{x\to\infty}(\sqrt{x + 9}-\sqrt{x + 4})\\)

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:

$0$