# Having trouble solving this limit

pyrosilver

## Homework Statement

lim (1+(a/x))^(bx) as x-->infinity

## The Attempt at a Solution

so, i raised the limit to e, and said e^lim(as x->inf) bxlog((a/x)+1). Then I pulled the constant b out and put it outside of the lim... I don't know how to do the rest though :( Was doing all my lim problems just fine until i came to this one

Mentor

## Homework Statement

lim (1+(a/x))^(bx) as x-->infinity

## The Attempt at a Solution

so, i raised the limit to e, and said e^lim(as x->inf) bxlog((a/x)+1). Then I pulled the constant b out and put it outside of the lim... I don't know how to do the rest though :( Was doing all my lim problems just fine until i came to this one

Raising things as a power of e is the wrong direction to go. Instead, let y = (1 + a/x)^(bx).

Now take the natural log of both sides, and then take the limit. You should get something you can use L'Hopital's rule on.

Homework Helper
Or, from where you left off, let y=a/x. Remember that this changes your limit from x approaches infinite to y approaches 0.

Homework Helper
Sorry, a rescaling would be much better. Let x=an

Last edited:
Staff Emeritus
Gold Member
Raising things as a power of e is the wrong direction to go. Instead ... Now take the natural log of both sides, and then take the limit.
That's more or less exactly what he did, he just organized the work differently.

Homework Helper
Do you know the power series expansion for log (1+x)? This will help you solve your problem.

Homework Helper
Do you know the power series expansion for log (1+x)? This will help you solve your problem.

I don't think that's really necessary. It would be assumed obvious to let $$\lim_{x\to \infty}\left(1+\frac{1}{x}\right)^{x}=e$$ so all that is required is to transform the question into a form that leaves this as the limit.

Homework Helper
I'd be uneasy saying that as e is defined using the natural numbers not the real numbers

Homework Helper
What does x approaching infinite have to do with either the natural or the real numbers?

Homework Helper
It still looks dodgy to me, I don't think you can say that. The route with expanding the log function is by far the safer route to go down.

gomunkul51
Here you go, the equivalent real value definition:

$$\lim_{x\to0}\left(1+x\right)^{\frac{1}{x}}=e$$

just don't use it all in one place! :)

Homework Helper
You then have to show that this in indeed the same value as the normal definition. No, I am convinced that there is far more work this way than my way.

gomunkul51