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

Find the limits:
$\underset{x\to \mathrm{\infty }}{lim}\left(\sqrt{x+9}-\sqrt{x+4}\right)$
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krolaniaN
Given:
The limits is $\underset{x\to \mathrm{\infty }}{lim}\left(\sqrt{x+9}-\sqrt{x+4}\right)$
Apply the limits.
$\underset{x\to \mathrm{\infty }}{lim}\left(\sqrt{x+9}-\sqrt{x+4}\right)=\mathrm{\infty }-\mathrm{\infty }$
This is an indeterminate for so multiply and divide the function with its conjugates and then apply the limits.
$\underset{x\to \mathrm{\infty }}{lim}\left[\left(\frac{\sqrt{x+9}-\sqrt{x+4}}{1}\right)\left(\frac{\sqrt{x+9}+\sqrt{x+4}}{\sqrt{x+9}+\sqrt{x+4}}\right)\right]=\underset{x\to \mathrm{\infty }}{lim}\left(\frac{\left(\sqrt{x+9}{\right)}^{2}-\left(\sqrt{x+4}{\right)}^{2}}{\sqrt{x+9}+\sqrt{x+4}}\right)$
$=\underset{x\to \mathrm{\infty }}{lim}\left(\frac{\left(x+9\right)-\left(x+4\right)}{\sqrt{x+9}+\sqrt{x+4}}\right)$
$=\underset{x\to \mathrm{\infty }}{lim}\left(\frac{5}{\sqrt{x+9}+\sqrt{x+4}}\right)$
$=\left(\frac{5}{\mathrm{\infty }}\right)$
$=0$
Result: $\underset{x\to \mathrm{\infty }}{lim}\left(\sqrt{x+9}-\sqrt{x+4}\right)=0$
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Jeffrey Jordon

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