Question

My homework assignment says: "Name all the ways a limit cannot exist." What exactly is it asking? Is it asking me to explain what a limit is? Or is it asking for when there isn't a limit?

Answer 1

For one, a limit doesn't exist when the left and right hand limits are not equal.

Answer 2

another way it could not exist is if you have a vertical asymptote at the limit point, i.e. the function goes to \(\infty\) or \(-\infty\)

Answer 3

@agent0smith how do the left and right hand limits not equal each other? Is that like with a one-sided limit? Like the graph f(x)=|x|/x?

Answer 4

And @satellite73 could you give an example function? I'm not quite sure what you mean

Answer 5

\[\lim_{x\to 2}\frac{x}{x-2}\] would be a simple example where the limit does not exist

Answer 6

The limit as you approach a point from the left or the right. Eg if the limit of a function as you approach a point from the left isn't the same as the limit when you approach the same point from the right.

Yes, just like f(x)=|x|/x the limit as you approach 0 from the left and right is not the same

Answer 7

btw \[f(x)=\frac{|x|}{x}\] is an example, but this is really a silly way to write
\[f(x) = \left\{\begin{array}{rcc}
-1 & \text{if} & x <0 \\
1& \text{if} & x >0
\end{array}
\right.
\]

Answer 8

as a piecewise function it is a lot easier to see why the limit does not exist at 0
because \(-1\neq 1\)

Answer 9

Yep, that's the first piecewise function that came to mind, for proving LHL ≠ RHL. But it was easier just to use the function he'd already posted, than write a piecewise function in latex :P