Draw a function $g$ such that $g$ is defined on the interval $[-9,9]$

Question

Sketch a function $g$ such that $g$ is defined on the interval $[-9,9]$ and satisfies the following properties:

a) $g(-5) = 1$, but $\lim_{x \to -5} g(x)$ does not exist. b) $\lim_{x \to 1} g(x) = 2$, but $g(1)$ can't equal $2$. c) $\lim_{x \to 0} g(x)$ does not exist.

I'm new to limits and graphing limits so any help would be appreciated.

Thanks!

Answer 1

The idea is to let go of the impression that graphics must be all nice and continuous. I'll give the ideas locally.

For a) think of the graphic of $\frac{1}{x}$ near zero. The function is not defined on zero, and the limit does not exist there. At the point $x = -5$, make something similar, make the lateral limits different, that is, the graph should, say, jump at this point. Like, a horizontal line ($\neq 1$) until it gets to $x = -5$, then make $g(-5) = 1$, and finally, another horizontal line after.

For b), the same idea for a) will do, take any graph, and make a point "jump" at $x = 1$.

For c), same as a).

Answer 2

This is very contrived, but here's a piece-wise function satsfying all of your conditions:

$g(x)=\begin{cases} 1 \ &\mbox{if} -9\leq x \leq -5 \\ \\ 1.5 \ &\mbox{if} -5<x<0 \\ \\ 2 \ &\mbox{if}\ 0 \leq x<1 \\ \\ 10 \ &\mbox{if} \ x=1 \\ \\ 2 \ &\mbox{if} \ 1<x \leq 9\end{cases}$

Your jobs:

• Draw the graph of this function
• Determine how the conditions that you've given ($a)$, $b)$ and $c)$) are satisfied in this function.