Logarithm limit

[wolly]

Hi,I want to find the limit of a limit:

$\lim_{x \to a} \frac{log_a[1+(x-a)]}{x-a}$

which comes from this limit:
$\lim_{x \to a} \frac{log_a[1+(x-a)]}{x^2-a^2}$
The answer to the first limit.is $(1)/ln(a)$
How did the limit reached that answer?
Can someone help?

 

[mario99]

Hi,I want to find the limit of a limit:

$\lim_{x \to a} \frac{log_a[1+(x-a)]}{x-a}$

which comes from this limit:
$\lim_{x \to a} \frac{log_a[1+(x-a)]}{x^2-a^2}$
The answer to the first limit.is $(1)/ln(a)$
How did the limit reached that answer?
Can someone help?

What have you done so far? Show your steps and I will correct them if you go wrong.

 

[Dr.Peterson]

Hi,I want to find the limit of a limit:

$$\lim_{x \to a} \frac{\log_a[1+(x-a)]}{x-a}$$

which comes from this limit:
$$\lim_{x \to a} \frac{\log_a[1+(x-a)]}{x^2-a^2}$$
The answer to the first limit.is $$(1)/\ln(a)$$
How did the limit reached that answer?
Can someone help?

You used the wrong delimiter for your Latex; a double dollar sign works, among other things. I corrected them above so I can read it.

What methods are available to you? Do you know any limits involving logs? Can you use L'Hopital's rule?

 

[mario99]

I just want to push the OP to try anything, so I will give him this Bonus idea. (Free Gift.)

$\displaystyle \lim_{x\rightarrow a}\frac{\log_a{[1 + (x - a)]}}{x - a} = \frac{1}{\ln a}\lim_{x\rightarrow a}\frac{\ln{[1 + (x - a)]}}{x - a}$

What methods are available to you? Do you know any limits involving logs? Can you use L'Hopital's rule?

If I were you (OP), I would just combine this Bonus with professor Dave's hint.

And

Thanks professor Dave for correcting the Latex.

 

[mario99]

Hi,I want to find the limit of a limit:

$\lim_{x \to a} \frac{log_a[1+(x-a)]}{x-a}$

which comes from this limit:
$\lim_{x \to a} \frac{log_a[1+(x-a)]}{x^2-a^2}$
The answer to the first limit.is $(1)/ln(a)$
How did the limit reached that answer?
Can someone help?

If you want to be fancy and solve the limit without L'hopital rule......

Hint: $\displaystyle \lim_{x \rightarrow 0}(x + 1)^{\frac{1}{x}} = e$.

 

[wolly]

I solved IT.Thanks!