Problems

01

Finding limits - indeterminate form $\frac{0}{0}$ - absolute value denominator

Find the limit: $$\underset{x\to -6}{\lim }\frac{2x+12}{|x+6|}$$

Hint: translate the absolute value function into a description using cases.

Explicitly mention all limit laws that you use.

02

Find limits at infinity

Find the limits:

• (a) $$\underset{x\to -\infty }{\lim }\frac{1-x^2}{x^3-x+1}$$
• (b) $$\underset{x\to +\infty }{\lim }\frac{x^4-3x^2+x}{x^3-x+2}$$

03

Find limit at infinity - algebra tricks

Find the limit: $$\underset{x\to -\infty }{\lim }\frac{x}{\sqrt{x^2+1}}$$

04

Choose parameter for continuity

For what values of the constant $c$ is the following function continuous for all $x$: $$h(x)=\{\begin{array}{ll}c-2x& \text{if }x\le 2\\ x^2-3x-c& \text{if }x>2\end{array}$$

05

Discontinuities

Consider the function: $$g(x)=\{\begin{array}{ll}{2}^x& x\le 1\\ 3-x& 1<x\le 4\\ \sqrt{x}& 4<x\end{array}$$

Where is $g(x)$ continuous?

For the points where $g(x)$ is discontinuous, determine whether it is continuous from the left and/or right.

06

Draw function with given limits

Draw a function which has $f(0)=0$ (crosses the origin) and the following limits:

• ${\lim }_{x\to -2^{+}}f(x)=\infty$
• ${\lim }_{x\to -2^{-}}f(x)=-\infty$
• ${\lim }_{x\to 2}f(x)=\infty$
• ${\lim }_{x\to -\infty }f(x)=0$
• ${\lim }_{x\to \infty }f(x)=1$