# Math Help - embarrasing algebra question

1. ## embarrasing algebra question

Hi,

In my calculus book i have the following limit problem:

$\lim_{x\to \infty}~\bigg(1+sin\frac{3}{x}\bigg)^x$

The first part of the solution:

1.Take the log of both sides:

$ln~y = \lim_{x\to \infty}~x~ln\bigg(1+sin\frac{3}{x}\bigg)$

Here's the part i don't understand:

$\frac{ln\bigg(1+sin\frac{3}{x}\bigg)}{\frac{1}{x}}$

Isn't dividing by 1/x the same as multiplying by x?

2. Originally Posted by Jones
Hi,

In my calculus book i have the following limit problem:

$\lim_{x\to \infty}~\bigg(1+sin\frac{3}{x}\bigg)^x$

The first part of the solution:

1.Take the log of both sides:

$ln~y = \lim_{x\to \infty}~x~ln\bigg(1+sin\frac{3}{x}\bigg)$

Here's the part i don't understand:

$\frac{ln\bigg(1+sin\frac{3}{x}\bigg)}{\frac{1}{x}}$

Isn't dividing by 1/x the same as multiplying by x?

Dear Jones,

Here the numerator and the denominator had been divided by x,

$xln\bigg(1+sin\frac{3}{x}\bigg)=\frac{xln\bigg(1+s in\frac{3}{x}\bigg)}{1}=\frac{ln\bigg(1+sin\frac{3 }{x}\bigg)}{\frac{1}{x}}$

Hope this will help you.

3. Originally Posted by Jones
Hi,

In my calculus book i have the following limit problem:

$\lim_{x\to \infty}~\bigg(1+sin\frac{3}{x}\bigg)^x$

The first part of the solution:

1.Take the log of both sides:

$ln~y = \lim_{x\to \infty}~x~ln\bigg(1+sin\frac{3}{x}\bigg)$

Here's the part i don't understand:

$\frac{ln\bigg(1+sin\frac{3}{x}\bigg)}{\frac{1}{x}}$

Isn't dividing by 1/x the same as multiplying by x?

Yes, it is- that the reason you can do this. Now, if you let y= 1/x, that becomes
$\lim_{y\to 0} \frac{ln(1+ sin(3y))}{y}$
That will be of the form "0/0" and you can use L'Hopital's rule.

4. Let y = (x/3) / (3/4). Y can be re-written as (x/3) * (4/3). Whenever you've got a rational expression, you can always multiply the numerator by the reciprocal of the denominator.

(sqrt(x)/2) / 2 = (sqrt(x)/2) * (1/2)

Adapted from wikipedia you have:

5. i think the OP knows that it can be written like that but just confused as to why you would do it. as hallsofivy said, you write it in that form so you can use L'hopital's rule.