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

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

Question
Limits and continuity
asked 2020-12-22
Find the limits:
\(\lim_{x\rightarrow\infty}(\sqrt{x+9}-\sqrt{x+4})\)

Answers (1)

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: \(\lim_{x\rightarrow\infty}(\sqrt{x+9}-\sqrt{x+4})=0\)
0

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