Question

Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it.
lim (1+a/x)^bx
x>infinite please i need help solving this and dont tell me do this step or this help me solve the problem all the way to the final answer

Answer 1

\[\Large\rm \lim_{x\to\infty}\left(1+\frac{a}{x}\right)^{bx}\]Hmm I think this relates to the definition of e.
But let's just work through it and see what's going on.

Answer 2

So right now we're getting the indeterminate form: \(\Large\rm 1^{\infty}\)
yes?

Answer 3

So there's a fancy little trick we can do.
\[\Large\rm e^{\ln \color{orangered}{x}}=\color{orangered}{x}\]This is true since the exponential and log are inverse functions of one another.
We want to use this but in reverse.\[\Large\rm \color{orangered}{\lim_{x\to\infty}\left(1+\frac{a}{x}\right)^{bx}}=e^{\ln\left[\color{orangered}{\lim_{x\to\infty}\left(1+\frac{a}{x}\right)^{bx}}\right]}\]

Answer 4

So let's ignore the base for a few steps...
and let's pass the limit out of the logarithm.
So we're looking at this:\[\Large\rm \lim_{x\to\infty} \ln\left[\left(1+\frac{a}{x}\right)^{bx}\right]\]

With me so far?
Where you at broski?

Answer 5

yes

Answer 6

Rules of logs allows us to bring the bx down in front,\[\Large\rm \lim_{x\to\infty} bx~ \ln\left(1+\frac{a}{x}\right)\]Let's take the b outside...
And we're going to do another fancy thing, this time with our x.
x is equal to the reciprocal of the reciprocal of x.
\[\Large\rm x=\frac{1}{\left(\frac{1}{x}\right)}\]The reason we would want to do this is,
it puts something in the denominator.
We want a numerator and denominator so we can check out L'Hopital maybe.

Answer 7

So here's what our stuff in the exponent looks like so far,
\[\Large\rm b\lim_{x\to\infty} \frac{\ln\left(1+\frac{a}{x}\right)}{\frac{1}{x}}\]If we take the limit now, what form are we getting?
(It should still be indeterminate, but what specific form?)

Answer 8

oh i see

Answer 9

when i had put that final answer into webassign it didnt accept ln(1+a/x)/1/x

Answer 10

correct. this simplifies down a lot further.
I was asking you a question.

Answer 11

oh okay

Answer 12

Now apply L'hospital's rule. Because they both are in 0/0 form.