Find Limit of Function: Homework Statement, Equations & Solution

[daveronan]

Homework Statement

Find the limit of the following function

lim w→∞ (1 + z/w)^w

Homework Equations

The Attempt at a Solution

lim w→∞ w ln(1 + z/w)

Not sure where to go next...

Thanks

 

[daveronan]

I've tried l'hospital's rule, but I don't think I'm any closer.

 

[A. Bahat]

Can you show how you tried l'Hopital's rule? Because I tried it and it worked.

 

[daveronan]

w/(1 + z/w) + 1/ln(1 +z/w)

 

[daveronan]

I think I may have forgot the chain rule...

 

[Dick]

w ln(1 + z/w) has the form infinity*0. You'll want to arrange it into an infinity/infinity form before you do l'Hopital. Can you show how you tried to apply it?

 

[daveronan]

w ln(1 + z/w) has the form infinity*0. You'll want to arrange it into an infinity/infinity form before you do l'Hopital. Can you show how you tried to apply it?

lim w→∞ w.ln(1/w*(w+z))

 

[daveronan]

After l'Hopital I'm getting lim w → ∞ ( -z/(w+z) + ln(1 + z/w))

This doesn't seem to help, not unless I'm doing something stupid.

 

[Dick]

lim w→∞ w.ln(1/w*(w+z))

That's still infinity*0. Try rearranging it into $\frac{ln(1+\frac{z}{w})}{\frac{1}{w}}$. That's 0/0.

 

[daveronan]

After l'Hopital I'm getting lim w → ∞ ( -z/(w+z) + ln(1 + z/w))

This doesn't seem to help, not unless I'm doing something stupid.

I've done that, but it still brings me to this.

 

[daveronan]

Wait, the penny dropped. The answer is z. You multiply across by -(w+z) :) Thanks for all your help.!

 

[Joffan]

... not forgetting you took the logarithm at the start.