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Math Help - Limits are limiting me

  1. #1
    Member M670's Avatar
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    Limits are limiting me

    Oh math Guru's I am just not getting how to find the limit of a composition with 2 graphs



    The blue is g(x) and red f(x)
    The graphs of and 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.
    2.
    3.
    4.
    #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?
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  2. #2
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    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)).
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  3. #3
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    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?

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  4. #4
    Member M670's Avatar
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    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  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....
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  5. #5
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    Re: Limits are limiting me

    Quote Originally Posted by M670 View Post
    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  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. \lim_{x\to 0^-} f(g(x))= 0
    Why are you confused about that?
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  6. #6
    Member M670's Avatar
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    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  x \rightarrow 3^- it would seem to be approximately 2 on the y axis
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  7. #7
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    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.

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