How Do You Solve the Limit of (1+(a/x))^(bx) as x Approaches Infinity?

[pyrosilver]

Homework Statement

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

Homework Equations

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

 

[Mark44]

Homework Statement

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

Homework Equations

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.

 

[Mentallic]

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

 

[Mentallic]

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

 

[Hurkyl]

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.

 

[hunt_mat]

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

 

[Mentallic]

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.

 

[hunt_mat]

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

 

[Mentallic]

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

 

[hunt_mat]

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! :)

 

[hunt_mat]

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]

@hunt_mat:

There are several equivalent definitions of e. it is not necessary to "prove" them because they are definitions, but it will be very interesting I agree :)

you can see the 4-5 common definitions at:
http://en.wikipedia.org/wiki/E_(mathematical_constant)