1. ## limits

evaluate lim as x goes to infinity (1+(a/x))^x

2. Originally Posted by Gigabyte

evaluate lim as x goes to infinity (1+(a/x))^x

Hint:

$\displaystyle \left(\lim_{x \to \infty} \left(1+\frac{a}{x}\right)^\frac{x}{a}\right)^a = e^a$

3. Hello,

Originally Posted by Gigabyte

evaluate lim as x goes to infinity (1+(a/x))^x
$\displaystyle \lim_{x \to \infty} {\color{red}\ln\left[\left(1+\frac ax\right)^x\right]}=\lim_{x \to \infty} x \ln \left(1+\frac ax\right)$

Change the variable : $\displaystyle t=\frac ax \implies x=\frac at$

When $\displaystyle x \to \infty$, $\displaystyle t \to 0$

$\displaystyle \lim_{x \to \infty} x \ln \left(1+\frac ax\right)=\lim_{t \to 0} \ln (1+t) \cdot \frac at=\lim_{t \to 0} a \cdot \frac{\ln(1+t)}{t}=a \cdot \lim_{t \to 0} \frac{\ln(1+t)-\ln(1)}{t}$

But $\displaystyle \lim_{t \to 0} \frac{\ln(1+t)-\ln(1)}{t}$ is, according to the definition of the derivative number, $\displaystyle \left(\ln(1+t)\right)'_{t=0}=\frac{1}{1+t} \bigg|_{t=0}=1$

Therefore, the limit is $\displaystyle \lim_{x \to \infty} {\color{red}\ln\left[\left(1+\frac ax\right)^x\right]}=a \cdot 1=\boxed{\color{red}a}$

But $\displaystyle \left(1+\frac ax\right)^x=e^{\color{red}\ln\left[\left(1+\frac ax\right)^x\right]}$

---> $\displaystyle \boxed{\lim_{x \to \infty} \left(1+\frac ax\right)^x=e^a}$

4. thanks heaps for that bru.

Is 1 line wrong though.
lim as t tends to 0

Isn't it supposed to be:
ln(1+t)^(a/t)

Not:
ln(1+t)*(a/t) as you wrote it??

5. Originally Posted by Gigabyte
thanks heaps for that bru.

Is 1 line wrong though.
lim as t tends to 0

Isn't it supposed to be:
ln(1+t)^(a/t)

Not:
ln(1+t)*(a/t) as you wrote it??
$\displaystyle \ln (a^b)=b \ln(a)$

6. yip that makes sense. Sorry one of those Rookie errors.

Why do you get?
(ln(1+t)-ln(1))/t
When you take (a) out the front of the limit. You take (a) out the front cos it's a constant right?

Then after that i kinda didn't get the next part.
"But is, according to the definition of the derivative number, "
Could you please explain that because i think i'm getting.

7. Originally Posted by Gigabyte
yip that makes sense. Sorry one of those Rookie errors.

Why do you get?
(ln(1+t)-ln(1))/t
$\displaystyle \ln(1)=0$, so I could add it, right ?

When you take (a) out the front of the limit. You take (a) out the front cos it's a constant right?
Yes

Then after that i kinda didn't get the next part.
"But is, according to the definition of the derivative number, "
Could you please explain that because i think i'm getting.
Have look here : Derivative - Wikipedia, the free encyclopedia

8. Originally Posted by Gigabyte
Then after that i kinda didn't get the next part.
"But is, according to the definition of the derivative number, "
Could you please explain that because i think i'm getting.

Are you introduced to differentiation? If not then you have to use limit definition of $\displaystyle \lim_{n \to \infty} \left(1+\frac1{n}\right)^n = e$

So follow what I said in post 2.

9. Originally Posted by Isomorphism
Are you introduced to differentiation? If not then you have to use limit definition of $\displaystyle \lim_{n \to \infty} \left(1+\frac1{n}\right)^n = e$

So follow what I said in post 2.
Is it possible that one is taught this limit before doing differentiation ?

LOL ! I forgot... by the way, his/her previous thread was about implicit differentiation

10. Originally Posted by Moo
Is it possible that one is taught this limit before doing differentiation ?
Ya obviously, We are taught limits first... how else will you learn the definition of differentiation?

Default
Quote:
Originally Posted by Isomorphism View Post
Are you introduced to differentiation? If not then you have to use limit definition of \lim_{n \to \infty} \left(1+\frac1{n}\right)^n = e

So follow what I said in post 2.
Is it possible that one is taught this limit before doing differentiation ?

LOL ! I forgot... by the way, his/her previous thread was about implicit differentiation
Oh ok then

11. Originally Posted by Isomorphism
Ya obviously, We are taught limits first... how else will you learn the definition of differentiation?
I don't see how you can prove this limit with tools that are taught before differentiation...

12. Yea i have been introduced to differentiation. However i don't really understand that line. I tried to read about it but got more confused.

So from post 8 would you get????
lim as x tents to infinity
ln[(1+a/x)^(x/a)]^a = e^a

13. Originally Posted by Gigabyte
Yea i have been introduced to differentiation. However i don't really understand that line. I tried to read about it but got more confused.
You just have to know that :

$\displaystyle \lim_{t \to 0} \frac{f(a+t)-f(a)}{t}=f'(a)$

Here, $\displaystyle a=1$ and $\displaystyle f(x)=\ln(1+t) \implies f'(x)=\frac{1}{1+t}$

So from post 8 would you get????
lim as x tents to infinity
ln[(1+a/x)^(x/a)]^a = e^a
yeah, this is the result

14. wow that was a mammoth step. But i think i have caught up to you. Now i'm stuck again lol.

What is the derivative number, and how does it help to solve the next line of working in post 3?? Otherwise i think i've got it.

Thank you so much for your help so far. It's been awesome, i'm really learning alot.

15. The derivative number in x=a is actually the value of the derivative at this point x=a.
It's the same as taking the value of the derivative function in a.

This is the very first definition of derivative that you should have learnt

Is it all clear now ?