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

[thegirl]

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.

 

[thegirl]

there's a better layout of the equation with x and y instead of w and wo

 

[Mark44]

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]

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
$$\frac{1}{\lim_{w \to w_0} \frac{e^w - e^{w_0}}{w - w_0}} $$

The denominator in the last expression is simply $\frac{d}{dw}e^w$ evaluated at $w=w_0$.