find the limit. (hint: try multiplying and dividing by the conjugate.)\n\\(\\lim_{x\\to+\\infty}(\\sqrt{x +…

Explanation:

Step1: Multiply and divide by conjugate

Multiply and divide $\lim_{x\rightarrow+\infty}(\sqrt{x + 36}-\sqrt{x + 81})$ by $\sqrt{x + 36}+\sqrt{x + 81}$: [ \begin{align*} &\lim_{x\rightarrow+\infty}\frac{(\sqrt{x + 36}-\sqrt{x + 81})(\sqrt{x + 36}+\sqrt{x + 81})}{\sqrt{x + 36}+\sqrt{x + 81}}\ =&\lim_{x\rightarrow+\infty}\frac{(x + 36)-(x + 81)}{\sqrt{x + 36}+\sqrt{x + 81}} \end{align*} ]

Step2: Simplify the numerator

Simplify the numerator $(x + 36)-(x + 81)$: [ \begin{align*} (x + 36)-(x + 81)&=x+36 - x-81\ &=- 45 \end{align*} ] So the limit becomes $\lim_{x\rightarrow+\infty}\frac{-45}{\sqrt{x + 36}+\sqrt{x + 81}}$.

Step3: Analyze the limit as $x\rightarrow+\infty$

As $x\rightarrow+\infty$, $\sqrt{x + 36}\rightarrow+\infty$ and $\sqrt{x + 81}\rightarrow+\infty$. Then $\sqrt{x + 36}+\sqrt{x + 81}\rightarrow+\infty$. We know that $\lim_{x\rightarrow+\infty}\frac{-45}{\sqrt{x + 36}+\sqrt{x + 81}} = 0$ since the numerator is a constant $-45$ and the denominator approaches infinity.

Answer:

$0$