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Homework Help: Limits problem

  1. Oct 19, 2011 #1
    1. The problem statement, all variables and given/known data
    Evaluate the following limit:-
    [tex]\lim_{x→0} \frac{e^{tan(x)}-e^x}{tan(x)-x}[/tex]

    2. Relevant equations

    3. The attempt at a solution
    Here are my attempts:-
    I firstly apply L'Hospital rule, i get:-
    [tex]\lim_{x→0} \frac{e^{tan(x)} \cdot sec^2(x)-e^x}{sec^2(x)}[/tex]

    If i directly substitute 0 here, i get the answer to be 0 but WolframAlpha says its 1. :confused:

    I don't understand where am i wrong? :(
  2. jcsd
  3. Oct 19, 2011 #2


    Staff: Mentor

    After applying L'Hopital's Rule, you're missing a term in the denominator.
    [tex]\lim_{x→0} \frac{e^{tan(x)} \cdot sec^2(x)-e^x}{sec^2(x) - 1}[/tex]

    This is still of the indeterminate form [0/0].
  4. Oct 19, 2011 #3
    Lol, sorry. :D

    Yeah, it is still of 0/0 form but when i again apply L'Hosiptal, it is still of 0/0 form. :(

    If i try to apply L'Hospital rule again and again, i am getting a 0 in denominator which will make my answer infinity, and the answer isn't that.

    Is there any other method?
  5. Oct 19, 2011 #4


    User Avatar
    Science Advisor
    Homework Helper

    The first and second derivatives the denominator are zero. The third derivative isn't.
  6. Oct 19, 2011 #5
    Yeah it isn't.

    Sorry again. I was doing some oral calculation, so i think i made a mistake. Sorry. :)

    EDIT: Thanks for all the help. I have solved the problem. :)
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