Functions and limits

[yobacul]

Can someone kindly explain this pls? I’m not understanding the way it’s written down. Kindly scroll down. Many thanks in advance.

 

[Dr.Peterson]

That is very poorly written! It hasn't defined g at all (probably due to a typo), and doesn't say what limit (f or g or something else) is requested (I guess both). The notation is also odd; the part to the right should be values of f(x), not mapping notation. I would have written the last one as

[MATH]g(x)=\left\{\begin{matrix} x^2,\;\;\text{if }x\ne 2\\ 0,\;\;\text{if }x=2\;\;\end{matrix}\right.[/MATH]​

Please show us the entire problem exactly as given.

Also, please tell us what you are thinking, so we can see where you are confused. What are the one-sided limits? What is required for the limit to exist?

 

[yobacul]

Thanks for your reply. This is the question exactly as given and I’m not understanding the way it’s written and it’s meaning. Thanks

 

[Dr.Peterson]

Thanks for your reply. This is the question exactly as given and I’m not understanding the way it’s written and it’s meaning. Thanks

Did my rewriting of it help? As I said, it is oddly written (and I was curious about the big blank space), but it should be clear the way I wrote it. (I didn't dare do f, because the formatting on this site doesn't handle these well.) If it didn't help, tell me what specifically you didn't understand, as I may be assuming your difficulty is in a different place.

Please answer the questions I asked, which are intended to lead you to the answer. Do you understand what it takes for a limit to exist, and how one-sided limits fit into the question? Is this from a textbook that gives examples of limits of piecewise functions? If not, see examples 4 and 5 here.

 

[pka]

Can someone kindly explain this pls? I’m not understanding the way it’s written down. Kindly scroll down. Many thanks in advance.

View attachment 22648

\(f(x)=\begin{cases}x\mapsto x^2 &\: x>2\\2\mapsto 2\\x\mapsto -x^2&\:x<2 \end{cases}\)

\(g(x)=\begin{cases}x\mapsto x^2 &\: x\ne 2\\2\mapsto 0\end{cases}\)
I realize that you say that what you posted was as it was given to you.
However, the above is what I think was really meant.
\(\mathop {\lim }\limits_{x \to {2^ + }} f(x) = 4~\&~\mathop {\lim }\limits_{x \to {2^ - }} f(x) =- 4\).
Therefore \(\mathop {\lim }\limits_{x \to 2} f(x) \text{ DNE}\)
However \(\mathop {\lim }\limits_{x \to 2} g(x) =4\)