ln 0 and Infinity

[Jason76]

The ln of 0 is infinity. Take this example:

The Limit as x approaches 0 from the right (positive side) of \(\displaystyle \dfrac{lnx}{x^{-1}}\)

So the top would be infinity as 0 is plugged in, but the bottom would be 0. So what is infinity divided by 0? But the book says the bottom is positive infinity, while the top is negative infinity, not just infinity. Why?

Once you actually use L'Hopital's rule then by taking the derivative you get a limit of 0.

 

[HallsofIvy]

The ln of 0 is infinity. Take this example:

No, the logarithm of 0 (to any base) does not exist. In terms of the limit we might say that ln(x) goes to negative infinity as x goes to 0.

The Limit as x approaches 0 from the right (positive side) of \(\displaystyle \dfrac{lnx}{x^{-1}}\)

So the top would be infinity as 0 is plugged in

No, it is not.

, but the bottom would be 0. So what is infinity divided by 0?

It's not anything. You cannot do arithmetic with "infinity" as if it were an actual number.

But the book says the bottom is positive infinity, while the top is negative infinity, not just infinity. Why?

What is ln(0.1)? What is ln(0.001)?

Once you actually use L'Hopital's rule then by taking the derivative you get a limit of 0.

L'Hopital's rule does not apply here. L'Hopital's rule only applies to "0/0" or "infinity/infinity"

 

[lookagain]

L'Hopital's rule does not apply here. L'Hopital's rule only applies to "0/0" or "infinity/infinity"

Jason76,

other indeterminate forms that sometimes can have L'Hopital's rule applied, with some
algebraic rewriting done first, are:

\(\displaystyle 1^{\infty}, \ \ \infty^0, \ \ \infty - \infty, \ \ (0)(\infty), \ \ and \ \ \ 0^0.\)

 

[HallsofIvy]

Jason76,

other indeterminate forms that sometimes can have L'Hopital's rule applied, with some
algebraic rewriting done first, are:

\(\displaystyle 1^{\infty}, \ \ \infty^0, \ \ \infty - \infty, \ \ (0)(\infty), \ \ and \ \ \ 0^0.\)

True and I thought about mentioning that. But L'Hopital's is applied, as you say "with some rewriting done first" to put them into the form "0/0" or "infinity/infinity".

Also, Jason76's example was clearly none of those forms. He seemed to think he was getting "infinity/infinity" but \(\displaystyle \lim_{x\to 0} x^{-1}\) is NOT "infinity".