# Finding the limit of lim w-->wo ((exp(w)-exp(wo))/(w-wo))^-1

Hi,

I know that when you take this limit it is equal to e^-wo, but I was just wondering how you got there when taking the limit?

lim w-->wo ((exp(w)-exp(wo))/(w-wo))^-1 = 1/e^wo

w and wo are both two points within the same plane.

Theres a better layout of the equation with x and y instead of w and wo

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Mark44
Mentor
Hi,

I know that when you take this limit it is equal to e^-wo, but I was just wondering how you got there when taking the limit?

lim w-->wo ((exp(w)-exp(wo))/(w-wo))^-1 = 1/e^wo

w and wo are both two points within the same plane.
You can evaluate the expression inside the outer parenthese using L'Hopital's Rule.

$$\lim_{w \to w_0} \left(\frac{e^w - e^{w_0}}{w - w_0} \right)^{-1}$$
The above is equal to
$$\frac{1}{\lim_{w \to w_0} \frac{e^w - e^{w_0}}{w - w_0}}$$

thegirl
Omg thank you so much!!!!!!!!!!!

mathman
$$\lim_{w \to w_0} \left(\frac{e^w - e^{w_0}}{w - w_0} \right)^{-1}$$
$$\frac{1}{\lim_{w \to w_0} \frac{e^w - e^{w_0}}{w - w_0}}$$
The denominator in the last expression is simply $\frac{d}{dw}e^w$ evaluated at $w=w_0$.