Question

Prove the following
\[
\lim_{x\to 0^+}x\ln(x)=0
\]

Answer 1

The simple brute force application of L'Hoptial rule won't work
\[
x\ln(x)=\frac{x}{\left(\ln(x)\right)^{-1}}\\
\lim_{x\to0^+}\frac{x}{\left(\ln(x)\right)^{-1}}=\lim_{x\to0^+}\frac{-1}{x\left(\ln(x)\right)^2}
\]

Answer 2

\[\lim_{x \rightarrow 0^+} \frac{\ln(x)}{\frac{1}{x}}=\lim_{x \rightarrow 0^+}\frac{\frac{1}{x}}{\frac{-1}{x^2}}=\lim_{x \rightarrow 0^+} -x=0\]

Answer 3

Really clever!

Answer 4

Is that really proving the limit is 0 though? Or is it just finding the limit?

Answer 5

what's the difference?

Answer 6

for example:
\[\lim_{x \rightarrow 2} \frac{x^2-4}{x-2}=4 \text{ we can find this limit algebraically } \\ \text{ but this isn't called a proof }\]

Answer 7

we can also find the limit using l'hospital too
but I'm also thinking that is not really considered a proof

Answer 8

sadly, it is

Answer 9

so i think

Answer 10

why do we need to prove the limit if we find it algebraically but we finding the limit using l'hospital means we have done the proof already..

Answer 11

http://math.stackexchange.com/questions/258273/how-do-i-prove-that-lim-x%E2%86%920-x%E2%8B%85-ln-x-0
this looks interesting

Answer 12

it looks like someone try to use squeeze theorem

Answer 13

the one with 5 votes

Answer 14

which makes sense to me

historically, limits came from trying to make sense of calculus, ie dividing by zero, which is what calculus does

so l'Hopital is calculus proving limits.

squeeze is geometry.

Answer 15

the one at the bottom is what occurred to me but i don't have the math.

Answer 16

I really like that squeeze theorem junk he did

Answer 17

Maybe if we look harder we can do this for all limits that exist

Answer 18

you know apply squeeze theorem
I know sometimes it might be tough

Answer 19

but he's integrating, so he's using limits to prove a limit 😰

Answer 20

I must eat
I might return

Answer 21

the sin x / x limit

that needs squeeze

that is the best example IMHO of where the circularity occurs.

Answer 22

I am asking a proof. \(\lim_{x\to0+}x\ln(x)=0\) is a statement which could be true or false. Therefore, you have to prove it. To express it as a find statement, I would have wrote the following

Find the following limit.
\[
\lim_{x\to0+}x\ln(x)
\]
It is pretty much a pointless distinction.

Answer 23

I thought squeeze theorem could be proved independently of the limit you are evaluating. I would not be surprised if you could derive it directly from the delta-epsilon definition of limits. @IrishBoy123

Answer 24

What is wrong with the integration and limit answer? He simply transformed a limit to a easier limit, just like we did using L'Hopital. Where is the circular reasoning?

I thought l'Hopital is calculus proving limit but then I checked the proof on wiki. Does not seem like this is the case.

https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule

Answer 25

For completeness,

https://en.wikipedia.org/wiki/Squeeze_theorem

Squeeze theorem can be proved using delta-epsilon alone.