Examples of calculus problems

Here is a collection of examples of calculus questions with hints, answers, solutions, common mistakes and other comments. You can click on the corresponding items to show/hide their content.

Limit with square root

Compute the limit. If the limit does not exist, determine with proof whether the trend is $\infty$, $-\infty$ or neither.
$\qquad\displaystyle\lim_{x\to 3}\frac{x-3}{\sqrt{3x}-3}$

Vertical asymptotes for function involving absolute value

Find all vertical asymptotes of the graph of the function
$\qquad\displaystyle f(x)=\frac{x^2+|x|-2}{x^2+x}$.
Justify your answer using limits.