How Do I Prove These Mathematical Limit Relations?

[mathmathRW]

I have been asked to prove the following limit relations.
(a) lim(as x goes to infinity) (b^x-1)/x = log(b)

(b) lim log(1+x)/x = 1

(c) lim (1+x)^(1/x) = e

(d) lim (1+x/n)^n =e^x

Unfortunately, I really have no idea where to start. We have a theorem that says if f(x)=the sum of (c sub n)*(x^n) then the limit of f(x) is the sum of c sub n. Is that useful for this problem? Any suggestions on how to do this?

 

[STEMucator]

How rigorous is your class? Do you prove things using the $\epsilon - \delta$ definition or are you simply trying to show that the limits are what they are?

 

[mathmathRW]

For this particular problem, I think we are just supposed to show that they are what they are.

 

[STEMucator]

For this particular problem, I think we are just supposed to show that they are what they are.

Let's try the first one : $lim_{x→∞} \frac{b^x-1}{x}$

Try plugging in some values and see what happens for x = 1, 2, 3... . That should draw your attention to what is happening in the numerator depending on $b$.

 

[HallsofIvy]

The "standard" rules of limits are not sufficient here. But you should think about this: what definition of "e" are you using?

 

[vela]

I have been asked to prove the following limit relations.
(a) lim(as x goes to infinity) (b^x-1)/x = log(b)

That can't be right. If b>1, the exponential function will grow much faster than $x$, so the limit will diverge.