Math Help - limit as x -> infty

1. limit as x -> infty

$\lim_{x \rightarrow \infty}(x-lnx)$

I made it into indeterminate form so that I can use l'Hospital's rule:
$\lim_{x \rightarrow \infty}(lne^x-lnx) = \lim_{x \rightarrow \infty}ln\left(\frac{e^x}{x}\right)$

I take the derivative and get:

$\frac{x-1}{x}$

Taking the derivative again I get that the function approaches 1, but based off of the actual graph the function approaches infinity. What am I doing wrong?

2. Re: limit as x -> infty

$\lim_{x\to\infty}\ln\left(\frac{e^x}{x} \right)=\ln\left(\lim_{x\to\infty}\frac{e^x}{x} \right)$

Now apply L'Hôpital's rule to the limit inside the log function.

3. Re: limit as x -> infty

Hello, amthomasjr!

$\lim_{x \rightarrow \infty}(x-\ln x)$

I made it into indeterminate form so that I can use l'Hospital's rule:

$\lim_{x\to\infty}(\ln e^x-\ln x) \:=\: \lim_{x\to\infty}\ln\left(\frac{e^x}{x}\right)$

I take the derivative and get: . $\frac{x-1}{x}$ . How?

Exactly what did you differentiate?

4. Re: limit as x -> infty

Originally Posted by Soroban
Hello, amthomasjr!

Exactly what did you differentiate?

$ln\left(\frac{e^x}{x}\right)$

5. Re: limit as x -> infty

In order to apply L'Hôpital's rule you need the limit to take the form:

$\lim_{x\to c}\frac{f(x)}{g(x)}$

6. Re: limit as x -> infty

Originally Posted by MarkFL2
$\lim_{x\to\infty}\ln\left(\frac{e^x}{x} \right)=\ln\left(\lim_{x\to\infty}\frac{e^x}{x} \right)$

Now apply L'Hôpital's rule to the limit inside the log function.
That makes sense, so

$ln\left(e^x\right) = x$ and

$\lim_{x\to\infty}x = \infty$

7. Re: limit as x -> infty

Originally Posted by MarkFL2
In order to apply L'Hôpital's rule you need the limit to take the form:

$\lim_{x\to c}\frac{f(x)}{g(x)}$
Which is why you only apply it to the inside, correct?

8. Re: limit as x -> infty

I would write:

$\ln\left(\lim_{x\to\infty}\frac{e^x}{x} \right)=\ln\left(\lim_{x\to\infty}e^x \right)=\ln(\infty)=\infty$

9. Re: limit as x -> infty

Originally Posted by amthomasjr
Which is why you only apply it to the inside, correct?
Right, on the inside of the log function, you now have the correct form to use the rule bought by L'Hôpital.

10. Re: limit as x -> infty

Originally Posted by MarkFL2
Right, on the inside of the log function, you now have the correct form to use the rule bought by L'Hôpital.
I'm assuming the same thing applies to trig functions if there were a similar case

11. Re: limit as x -> infty

Yes, I believe so as long as the trig function is continuous.