9. find the limit.\n\\(\\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\rightarrow\infty}(\sqrt{x + 9}-\sqrt{x + 4})\times\frac{\sqrt{x + 9}+\sqrt{x + 4}}{\sqrt{x + 9}+\sqrt{x + 4}}\ =&\lim_{x\rightarrow\infty}\frac{(\sqrt{x + 9})^2-(\sqrt{x + 4})^2}{\sqrt{x + 9}+\sqrt{x + 4}}\ =&\lim_{x\rightarrow\infty}\frac{(x + 9)-(x + 4)}{\sqrt{x + 9}+\sqrt{x + 4}}\ =&\lim_{x\rightarrow\infty}\frac{5}{\sqrt{x + 9}+\sqrt{x + 4}} \end{align*} ]

Step2: Analyze the limit as $x\rightarrow\infty$

As $x\rightarrow\infty$, both $\sqrt{x + 9}\rightarrow\infty$ and $\sqrt{x + 4}\rightarrow\infty$. So, $\lim_{x\rightarrow\infty}\frac{5}{\sqrt{x + 9}+\sqrt{x + 4}} = 0$.

Answer:

$0$