Find the limit if it exists

I'm having a really difficult time understanding limits from the graph. I don't completely understand how to find if the limit exists or not. I tried to do these for my homework but I don't know if they are right. Could someone please help me understand limits?

http://i45.tinypic.com/152hn36.jpg

Re: Find the limit if it exists

B and M are incorrect. Can you recheck those and see what the problem is?

Also, the graph is a bit unclear as x -> 4... is that supposed to be an asymptote, or do the curves actually stop at $\displaystyle \pm5$? If theres an asymptote, the one sided limits will be $\displaystyle \pm\infty$, the regular two-sided limit won't exist. But I think that's an asymptote, so F and G are wrong too.

To answer your question, a one sided limit will exist if the function approaches a particular value (or to infinity), and gets arbitrarily close as you get closer and closer, FROM that direction.

The usual two-sided limit will exist ONLY if the left-hand and right-hand limits exist and are equal.

The function will be continuous if the limit exists, AND the limit actually equals the function value. Notice that this does actually happen at x=6.

Edit: A looks incorrect as well.

Re: Find the limit if it exists

By my count, you have 4 incorrect.

Re: Find the limit if it exists

Hello, calculus123!

Most of your answers are correct . . . good work!

Quote:

I'm having a really difficult time understanding limits from the graph.

I don't completely understand how to find if the limit exists or not.

I tried to do these for my homework but I don't know if they are right.

Could someone please help me understand limits?

http://i45.tinypic.com/152hn36.jpg

Your answer for (a) is incorrect.

Trace the graph approaching -3 from the left.

. . The graph approaches $\displaystyle f(x) =1.$

Trace the graph approaching -3 from the right.

. . The graph approaches $\displaystyle f(x) = 1.$

Therefore: .$\displaystyle \lim_{x\to\text{-}3}f(x) \;=\;1$

Your answer for (g) is off.

It looks like it goes to 5,

. . but I believe there is a vertical asymptote at $\displaystyle x = 4.$

Therefore: .$\displaystyle \lim_{x\to4^+}}f(x) \:=\:+\infty \quad\text{ and }\,\lim_{x\to4^-}f(x) \:=\:-\infty$

Your answer to (m) is incorrect.

$\displaystyle f(x)$ *is* continuous at $\displaystyle x = 6.$

(The graph doesn't "break" there, does it?)

Re: Find the limit if it exists

Thank you Soroban and SworD! I seem to understand them a little bit better now. I made all the changes that you guys suggested, are they correct now? I tried to redo them and they seem to make more sense now. Thanks!

http://i50.tinypic.com/33k4ya1.jpg

Re: Find the limit if it exists

Only B is still incorrect. Notice that they want the limit from the right.

Re: Find the limit if it exists

So the answer to b would be 2 since it's coming from the right?

Re: Find the limit if it exists

Yes. Limits don't care at all about the value *at* that point.

Re: Find the limit if it exists

Quote:

Originally Posted by

**SworD** Only B is still incorrect. Notice that they want the limit from the right.

**That in not correct.**

Quote:

Originally Posted by

**calculus123** So the answer to b would be 2 since it's coming from the right?

**That in not correct.**

b) $\displaystyle {\lim _{x \to {2^ + }}}f = 1$.

Re: Find the limit if it exists

For b), It's $\displaystyle {\lim _{x \to {-2^ + }}}f = 1$, **not** $\displaystyle {\lim _{x \to {2^ + }}}f = 1$. Are you sure the answer is 1 and not 2? I'm looking at the graph again and it seems like the answer is 2.

Re: Find the limit if it exists

Quote:

Originally Posted by

**calculus123** For b), It's $\displaystyle {\lim _{x \to {-2^ + }}}f = 1$, **not** $\displaystyle {\lim _{x \to {2^ + }}}f = 1$. Are you sure the answer is 1 and not 2? I'm looking at the graph again and it seems like the answer is 2.

The image you posted is very hard to read.

You should try to make questions clear to the reader.

Re: Find the limit if it exists

It's pretty clear that the number being approached in (b) is -2... unless they decided to make the -> arrow twice as big and have a wedge drawn on it.

Re: Find the limit if it exists

Quote:

Originally Posted by

**SworD** It's pretty clear that the number being approached in (b) is -2... unless they decided to make the -> arrow twice as big and have a wedge drawn on it.

It may be different on different computer screens. On mine it appears $\displaystyle \to 2$ as opposed to $\displaystyle \to -2$.

I cannot understand why any serious user of this site does not learn to use LaTeX.

Re: Find the limit if it exists

Quote:

Originally Posted by

**Plato** It may be different on different computer screens. On mine it appears $\displaystyle \to 2$ as opposed to $\displaystyle \to -2$.

I cannot understand why any serious user of this site does not learn to use LaTeX.

I don't blame you for not seeing the negative sign because I have the sheet in front of me printed and I can't see it either. Thanks for your help!