Exercises

01

One-sided limits

Find the limits:

• (a) ${\lim }_{x\to 5^{+}}\frac{x+1}{x-5}$
• (b) ${\lim }_{x\to 3^{-}}\frac{\sqrt{x}}{(x-3{)}^5}$

02

Rational function - limits and asymptotes

Consider the function: $$f(x)=\frac{x^2-9x+14}{x^2-5x+6}$$

This function is undefined for two values of $x$. For each such value, find the right, left, and total limits of $f(x)$ as $x$ approaches that value. What are the vertical asymptotes of $f$?

What are the horizontal asymptote(s) of $f$?

03

Finding limits - indeterminate form $\frac{0}{0}$

Find the limits:

• (a) $$\underset{x\to 6}{\lim }\frac{x-6}{x^2-36}$$
• (b) $$\underset{x\to 0}{\lim }\frac{\frac{1}{5+x}-\frac{1}{5}}{x}$$

Explicitly mention the limit laws that you use.

04

A removable continuity?

Here is a definition of continuity: $f(x)$ is continuous at $x=a$ when ${\lim }_{x\to a}f(x)=f(a)$.

Is this function $f(x)$ continuous at $x=1$?: $$f(x)=\{\begin{array}{ll}\frac{x^2-x}{x^2-1}& \text{if }x\ne 1\\ 1& \text{if }x=1\end{array}$$