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Limits of Functions

By Rationalization

         

Evaluate \(\displaystyle \lim_{x \to -\infty} \frac{24\sqrt{14}x}{29-\sqrt{14x^2+56}} \).

Evaluate \[\lim_{x \to \infty} \left(\sqrt{x+18}-\sqrt{x}\right).\]

Evaluate \(\displaystyle \lim_{x \to 0} \frac{224\left(\sqrt{16+x^2}-4 \right)}{x^2}\).

Evaluate

\[ \displaystyle \lim_{x \to 999^+} \tan^{-1} \left(\frac{1}{x - 999}\right). \]

Details and assumptions

\(\tan^{-1}x\) denotes the inverse of \(\tan x\) and not the reciprocal \(\frac{1}{\tan x}\).

The principal branch of \( \tan^{-1}\) is \(\left( -\frac{\pi}{2}, \frac{\pi}{2}\right)\).

Evaluate \(\displaystyle \lim_{x \to -\infty} \left(\frac{-17x}{\sqrt{x^2+17}} + 24\right)\)

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