# Limits are limiting me

• Jan 26th 2013, 02:37 PM
M670
Limits are limiting me
Oh math Guru's I am just not getting how to find the limit of a composition with 2 graphs

http://webwork.mathstat.concordia.ca...rob5image2.pnghttp://webwork.mathstat.concordia.ca...rob5image1.png

The blue is g(x) and red f(x)
The graphs of http://webwork.mathstat.concordia.ca...910e13a9c1.png and http://webwork.mathstat.concordia.ca...b5d4817b11.png are given above. Use them to evaluate each quantity below. Write DNE if the limit or value does not exist (or if it's infinity).

1. http://webwork.mathstat.concordia.ca...6be18b0341.png
2. http://webwork.mathstat.concordia.ca...ebc2b21421.png
3. http://webwork.mathstat.concordia.ca...075e1739e1.png
4. http://webwork.mathstat.concordia.ca...679c592751.png
#2 and 3 are correct(#2=4 and #3=4) but I can't figure out 1 and 4 I thought #1 was 3 and #4 was 1
I understand where to look on the graph, I just don't understand how to relate the points to one another?
• Jan 26th 2013, 06:12 PM
chiro
Re: Limits are limiting me
Hey M670.

Hint: You might want to consider a substitution where lim u -> g(x)- (i.e. from the left) and consider this in terms of f(u) instead of f(g(x)).
• Jan 26th 2013, 08:35 PM
hollywood
Re: Limits are limiting me
For #1, you can find the limit of g(x) and just look at the graph to see what f(g(x)) is there.

For #4, you need to think a little harder. Since x goes to 3 from above, g(x) goes to 0 from above. Can you see that in the graph? So now you look at the graph for f, and when its input goes to 0 from above, what does its output do?

- Hollywood
• Jan 27th 2013, 05:56 AM
M670
Re: Limits are limiting me
Oh I think I understand what I need to do, if I treat this like when doing compositions of function I figure out my g(x) and input it as my (x) in the f(g(x))
Ok but for #1 g(x) I am looking at the blue graph and my $\displaystyle x\rightarrow 0^-$ which means I am looking from right to left, this is where I get confused cuz it looks like its around 1 but then x @ 1 in the red graph gives us 0 on the y axis....
• Jan 27th 2013, 06:01 AM
HallsofIvy
Re: Limits are limiting me
Quote:

Originally Posted by M670
Oh I think I understand what I need to do, if I treat this like when doing compositions of function I figure out my g(x) and input it as my (x) in the f(g(x))
Ok but for #1 g(x) I am looking at the blue graph and my $\displaystyle x\rightarrow 0^-$ which means I am looking from right to left, this is where I get confused cuz it looks like its around 1 but then x @ 1 in the red graph gives us 0 on the y axis....

Yes, in #1, as x goes to 0 g(x) goes to 1. And then, as g(x) goes to 1, f(g(x)) goes to 0. $\displaystyle \lim_{x\to 0^-} f(g(x))= 0$
Why are you confused about that?
• Jan 27th 2013, 06:19 AM
M670
Re: Limits are limiting me
Ok so then for #4 g(x) goes to 0 and the f(x) goes to 4 is my first thought but looking at $\displaystyle x \rightarrow 3^-$ it would seem to be approximately 2 on the y axis
• Jan 27th 2013, 06:34 AM
hollywood
Re: Limits are limiting me
Yes, that's right: x goes to 3 from above, so g(x) goes to 0 from above, since the graph of g(x) is above the x-axis just to the right of x=3. Since g(x) goes to 0 from above, f(g(x)) goes to 2.

- Hollywood