- #1
robertdeniro
- 39
- 0
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:
I 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.
You added "Show f(0) = 1/2" but you haven't defined f(x).
fixed that too
OK, 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.
I 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.
Write 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.
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
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