# Find the limits: lim_{xrightarrowinfty}(sqrt{x+9}-sqrt{x+4})

Question
Limits and continuity
Find the limits:
$$\lim_{x\rightarrow\infty}(\sqrt{x+9}-\sqrt{x+4})$$

2020-12-23
Given:
The limits is $$\lim_{x\rightarrow\infty}(\sqrt{x+9}-\sqrt{x+4})$$
Apply the limits.
$$\lim_{x\rightarrow\infty}(\sqrt{x+9}-\sqrt{x+4})=\infty-\infty$$
This is an indeterminate for so multiply and divide the function with its conjugates and then apply the limits.
$$\lim_{x\rightarrow\infty}[(\frac{\sqrt{x+9}-\sqrt{x+4}}{1})(\frac{\sqrt{x+9}+\sqrt{x+4}}{\sqrt{x+9}+\sqrt{x+4}})]=\lim_{x\rightarrow\infty}(\frac{(\sqrt{x+9})^2-(\sqrt{x+4})^2}{\sqrt{x+9}+\sqrt{x+4}})$$
$$=\lim_{x\rightarrow\infty}(\frac{(x+9)-(x+4)}{\sqrt{x+9}+\sqrt{x+4}})$$
$$=\lim_{x\rightarrow\infty}(\frac{5}{\sqrt{x+9}+\sqrt{x+4}})$$
$$=(\frac{5}{\infty})$$
$$=0$$
Result: $$\lim_{x\rightarrow\infty}(\sqrt{x+9}-\sqrt{x+4})=0$$

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