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Homework Statement
[tex]f(x)=\frac{e^{x}-e^{-\sqrt{x}}}{e^{\sqrt{x}}-e^{-\sqrt{x}}}[/tex]
show f(0)=1/2
Homework Equations
The Attempt at a Solution
Last edited:
hey, sorry its fixed nowI think you have a sign wrong somewhere, otherwise the LHS = 1. But if you can express your equation in the form
[tex] e^{\sqrt{x}} = A e^x[/tex]
you can take a logarithm of both sides to get a more manageble equation to solve.
fixed that tooYou added "Show f(0) = 1/2" but you haven't defined f(x).
OK, the image probably didn't reload in my browser.fixed that too
yes i tried lopital twice, didnt work and it was getting really messy so i stoppedOK, the image probably didn't reload in my browser.
Have you tried taking the limit as [tex]x\rightarrow 0[/tex]? You should really show some of your work before asking for help.
sorry im not followingI think it helps to note that you can rewrite
[tex]
f(x)=\frac{e^{x+\sqrt{x}}-1}{e^{2\sqrt{x}}-1}.
[/tex]
Then you can factor the denominator and you get an expression where you can use L'Hopital on an indeterminant factor.
ok i seeWrite that as a product of two factors, one of which is well-defined in the limit and the other which is indeterminate. L'Hopital can be applied to the indeterminate factor.
If the function is undefined at a point, then we should attempt to define it by it's limit at that point, if such limit exists.ok i see
but we evaluate its limit going to 0, but not AT 0.
i do not recall a theorem that says limit=actual value
There are 3 ways you can go about solving this problems:Homework Statement
[tex]f(x)=\frac{e^{x}-e^{-\sqrt{x}}}{e^{\sqrt{x}}-e^{-\sqrt{x}}}[/tex]
show f(0)=1/2
Homework Equations
The Attempt at a Solution