Homework Help: Natural log

1. Apr 13, 2005

UrbanXrisis

$$lim _{x->1} \frac{x}{ln x}$$

L'Hopital's rule:
$$lim _{x->1} \frac{1}{1/x}$$

so the limit is 1

is this correct?

2. Apr 13, 2005

SpaceTiger

Staff Emeritus
Since the numerator goes to 1 in the first expression, you shouldn't need l'Hopital's rule, you can just say the limit is infinity.

3. Apr 13, 2005

shmoe

It's actually incorrect to apply l'hopital here, it's not in an indeterminate form.

I wouldn't say the limit is infinity either- examine the left and right hand limts seperately.

4. Apr 13, 2005

UrbanXrisis

ln(1) = zero, so the function is undefined, wouldn't that make L'Hopotal's rul valid here?

I graphed this and it gave the limit doesn't exist. I'm not sure how I would show this by my calculations

5. Apr 13, 2005

dextercioby

The left limit is -infinity,while the right one is +infinity.

Daniel.

P.S.It's not an indeterminate form,because the numerator goes to 1,while the denominator goes to 0.

6. Apr 13, 2005

UrbanXrisis

how do you prove that the left limit is -infinity,while the right one is +infinity without a calc?

7. Apr 13, 2005

dextercioby

$$\lim_{x\nearrow 1}\frac{1}{\ln x} =\frac{1}{0^{-}}=-\infty$$

$$\lim_{x\searrow 1}\frac{1}{\ln x} =\frac{1}{0^{+}}=+\infty$$

Daniel.

8. Apr 13, 2005

Hurkyl

Staff Emeritus
It's not too hard when the values are strictly negative on the left and strictly positive on the right.

9. Apr 13, 2005

jdavel

UrbanXrisis,

Try drawing a graph of ln(x).

10. Apr 13, 2005

UrbanXrisis

how did you get negative zero and positive zero?

11. Apr 13, 2005

dextercioby

From the values of $\ln x$...?One is approaching zero from below and the other from above,hence the notation.

Daniel.

12. Apr 13, 2005

UrbanXrisis

but how do you know that without a calculator? I mean, if you subbed in 1, you would just get ln(1)=0 not +0 or -0

13. Apr 13, 2005

Hurkyl

Staff Emeritus
Do you recall the definition of a one-sided limit?

14. Apr 13, 2005

UrbanXrisis

no, I forgot, please remind me

15. Apr 13, 2005

Hurkyl

Staff Emeritus
You could always check your text -- learning how to get information from your textbook is an important skill. :tongue:

Seriously, though: you should be able to reread the good introduction in your book quicker than it would take any of us to write a good introduction... it would probably be better too. If you still have problems with it, you can come back and ask for clarification!

16. Apr 13, 2005

UrbanXrisis

we dont have text books... all we have are the notes we take in class, no joke. The only way we were taught was to fraph the function. However, what if we did not have a calculator?

17. Apr 13, 2005

jdavel

UrbanXrisis,

"The only way we were taught was to graph the function. However, what if we did not have a calculator?"

Yikes!

I don't know what you're planning to do in your life, but if it has anything to do with science or math, having to rely on a calculator to draw a graph of ln(x) will be as crippling as having to rely on it to calculate 2+2. You can't build knowledge when your entire foundation resides in a calculator.

18. Apr 13, 2005

UrbanXrisis

that is very correct, I did not mean it that literally, it was just to get someone to post the non-graphing way of calculating the limit from the left and from the right. It's not in my notes