Is the Calculation of lim_{x->1} (x/ln x) Using L'Hopital's Rule Correct?

[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?

 

[SpaceTiger]

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.

 

[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.

 

[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

 

[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.

 

[UrbanXrisis]

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

 

[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.

 

[Hurkyl]

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

 

[jdavel]

UrbanXrisis,

Try drawing a graph of ln(x).

 

[UrbanXrisis]

\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.

how did you get negative zero and positive zero?

 

[dextercioby]

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

Daniel.

 

[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

 

[Hurkyl]

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

 

[UrbanXrisis]

no, I forgot, please remind me

 

[Hurkyl]

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

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!

 

[UrbanXrisis]

we don't have textbooks... 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?

 

[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.

 

[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