Evaluate Limit: [tex]\lim_{T\rightarrow 0^+} \ln(2e^{-\epsilon/kT}+1)[/itex]

[quasar987]

I'm having a blank here. How do I evaluate

[tex]\lim_{T\rightarrow 0^+} \ln(2e^{-\epsilon/kT}+1) \ \ \ \ \ \ \ (\epsilon>0, \ \ k>0)[/itex]

?!?

 

[DeadWolfe]

Well, e^-(1/x) tends to 0 with x to 0, so our whole expression tends to log(0+1)=0. (Of course, we need k and epsilon positive, and we really should take a right-handed limit, since for negative T the expression blows up with T to 0.)

 

[quasar987]

What justifies your moving the limit inside the log?

(I refined the expresion following your comments, thx)

 

[slearch]

Continuity of ln away from 0.

 

[quasar987]

Oh I have it. It's that

$$\lim_{x\rightarrow x_0} (f\circ g)(x)$$

will equal

$$f(\lim_{x\rightarrow x_0} g(x))$$

if the limit of g exists and it f is continuous as $\lim_{x\rightarrow x_0} g(x)$.

 

[DeadWolfe]

Most textbooks define continuity this way.