×

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

×