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How do you solve limits with f notations in them?

  1. Dec 6, 2016 #1
    1. The problem statement, all variables and given/known data
    Problem 1:
    ##f'(1) = -2##
    Solve:
    $$\lim_{x\to0} \frac{f(e^{5x} - x^2) - f(1)}{x}$$


    2. Relevant equations


    3. The attempt at a solution
    Okay so these type of problems really get to me. I'm going to assume some level of substitution are needed but I'm really unsure.

    I'm guessing that I can do something like ## u = e^5x - x^2##(or maybe i have to do something like## h(x) = f(e^{5x} - x^2)##.

    But what would I do from there? Would f(1) be equal to h(1)? Do I just use l'hopitals if this is on the right path? Is there a standard procedure to follow when dealing with limits that have function notations in them?
     
    Last edited: Dec 6, 2016
  2. jcsd
  3. Dec 6, 2016 #2

    Math_QED

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    Is this all the information you got? How would you apply Hopital when you don't know that the numerator goes to zero?
     
  4. Dec 6, 2016 #3

    PeroK

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    Is that supposed to be ##e^{5x}##?
     
  5. Dec 6, 2016 #4

    Mark44

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    Are you sure it isn't ##f(e^{5x} - x^2)##?
     
  6. Dec 6, 2016 #5

    vela

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    That limit looks to me suspiciously similar to a derivative. If you use ## u = e^{5x} - x^2##, you'd have f(u)-f(1) in the numerator. What would you need in the denominator to get the derivative f'(1)?
     
  7. Dec 6, 2016 #6
    Yeah this is all the information I have. I forgot that L'hopitals can only be used if indeterminate form is found.
    Yeah you're right. Sorry about that i'll edit it
    Are you saying that x should be equal to f'(1)?
     
  8. Dec 6, 2016 #7

    vela

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    No. I'm not sure how you jumped to that conclusion.
     
  9. Dec 6, 2016 #8

    Stephen Tashi

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    You should be able to show that the limit of the numerator is zero and this will justify using l'hospitals rule. . ( i.e. show ##lim_{x \rightarrow 0} {f(e^{5x} - x^2) - f(1)} = f(1)-f(1) = 0## ) To show that, you need to use the fact that if f'(x) exists at x = a then f is continuous at x = a and also you need a result that tells about the limit of a composition of two functions. There is probably a theorem in your text materials that tells some conditions which imply that ##lim_{x \rightarrow a} f(g(x)) = f(g(a))##.
     
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