Don't Do This

Incorrect: $\lim_{x\to\infty}\frac{5}{x}=\frac{5}{\infty}=0$
Correct: $\lim_{x\to\infty}\frac{5}{x}=0$

Incorrect: $\lim_{x\to\infty}\frac{x^2}{x}=\frac{\infty}{\infty}$ Undefined.
Correct: $\lim_{x\to\infty}\frac{x^2}{x}=\lim_{x\to\infty}x=\infty$

Incorrect: $\lim_{x\to\infty}\frac{x}{x^2}=\frac{\infty}{\infty}$ Undefined.
Correct: $\lim_{x\to\infty}\frac{x}{x^2}=\lim_{x\to\infty}\frac{1}{x}=0$

Incorrect: $\lim_{x\to\infty}\frac{\ln x}{4x}=\frac{\infty}{\infty}$ Undefined.
Correct: $\lim_{x\to\infty}\frac{\ln x}{4x}=\lim_{x\to\infty}\frac{1/x}{4}=0$

Incorrect: $\lim_{x\to 0}\frac{\sqrt{x}}{6x}=\frac{0}{0}$ Undefined.
Correct: $\lim_{x\to 0}\frac{\sqrt {x}}{6x}=\lim_{x\to\infty}\frac{1}{6\sqrt{x}}=\infty$

Incorrect: $\lim_{x\to \infty}\frac{\sqrt{x}}{6x}=\frac{\infty}{\infty}$ Undefined.
Correct: $\lim_{x\to \infty}\frac{\sqrt {x}}{6x}=\lim_{x\to\infty}\frac{1}{6\sqrt{x}}=0$

Incorrect: $\lim_{x\to \infty}\left(\sqrt{x}-\sqrt{x+1} \right)=\infty-\infty=0$
Correct: $\lim_{x\to \infty}\left(\sqrt{x}-\sqrt{x+1} \right)=\lim_{x\to \infty}\frac{-1}{\sqrt{x}+\sqrt{x+1}}=0$

Incorrect: $\lim_{x\to \infty}\left(\sqrt{x}-\sqrt{4x} \right)=\infty-\infty=0$
Correct: $\lim_{x\to \infty} \left(\sqrt{x}-\sqrt{4x} \right)=\lim_{x\to \infty}\frac{-3x}{\sqrt{x}+\sqrt{x+1} }\overset{\mbox{L'H}}{=}\lim_{x\to\infty} \frac{-3}{\frac{1}{2\sqrt{x}}+\frac{1}{2\sqrt{x+1}}}=\infty$
Correct: $\lim_{x\to \infty}\left(\sqrt{x}-\sqrt{4x} \right)=\lim_{x\to \infty}\frac{-3x}{\sqrt{x}+\sqrt{x+1}}$
$= \lim_{x\to \infty}\frac{-3x}{\sqrt{x}+\sqrt{x+1}}\cdot\frac{1/\sqrt{x}}{1/\sqrt{x}}=\lim_{x\to \infty}\frac{-3\sqrt{x}}{1+\sqrt{1+1/x}}=\infty$