Find Limit of Function: tanhx-x/x as x→0

[wolfspirit]

Hi,

I'm trying to find the limit as x tends to zero of the function (tanhx-x)/x
this is what i have but i have no idea if i am on the right lines?
lim x-> 0 can be split up into two problems:
limx->0 (tanhx)/x - limx->0 x/x
limx->0 x/x = 1
limx->0 (tanhx)/x can be expressed as
limx->0 ((e^x-e^-x)/e^x+e^-x))/x
=
limx->0 (x(e^x-e^-x)/e^x+e^-x)) =0
which leave the limit as -1 but wolfram gives a limit of 0

how should i be approaching this problem?
many thanks
Ryan

 

[BiGyElLoWhAt]

in your final expression, I'm not sure how you're getting -1
[e^0 - e^(-0)] =? I think you plugged in your limit wrongly.

Edit: Oh wait, I thought you were saying that the limit of tanh was -1.
Your problem lies in the fact that $\frac{tanh(x)}{x} \neq \frac{x(e^x-e^{-x}}{e^x +e^{-x})}$

$\frac{tanh(x)}{x} = \frac{e^x-e^{-x}}{x(e^x +e^{-x})}$

 

[wolfspirit]

With the x/x you can use l'hospital's rule to get round the fact that you have somepthing that is tending to 0/0 but for the tanh(x)/x part you cant? so how do you evaluate it? and I am a bit confused as to how

tanh(x)x=ex−e(−xx(ex+e(−x))

Because i thought
((e^x-e^-x)/e^x+e^-x))/x was the same as:
((e^x-e^-x)/e^x+e^-x)) times x/1 ?many thanks for your reply :)
Ryan

 

[BiGyElLoWhAt]

You can use the hospital rule. But you didn't express tanh x/x correctly.
$\frac{tanh x}{x} = \frac{1}{x}tanh x = \frac{1}{x}\frac{e^x - e^{-x}}{e^x + e^{-x}} = \frac{e^x-e^{-x}}{x(e^x + e^{-x})}$

 

[BiGyElLoWhAt]

In your case, you end up with a 0/1, but in this case you don't.

 

[wolfspirit]

ah brilliant much clearer thanks! :)

 

[BiGyElLoWhAt]

Not a problem.