Question

Solve : $\lim _{x\to \:0}\left(x\ln \left(x\right)\right)$

Answer 1

$\lim _{x\to \:0}\left(x\ln \left(x\right)\right)=0$

$Steps$

$\mathrm{If\:}\lim _{x\to a-}f\left(x\right)=\lim _{x\to a+}f\left(x\right)=L\mathrm{\:then}\:\lim _{x\to a}f\left(x\right)=L$

$\lim _{x\to \:0-}\left(x\ln \left(x\right)\right)=0$

$\mathrm{Indeterminate\:form:}\:\infty \cdot 0$

$\lim _{x\to \:0+}\left(x\ln \left(x\right)\right)=0$

$\mathrm{Indeterminate\:form:}\:\infty \cdot 0$

= 0

Answer 2

Explanation:

limx→0(xln(x)+1)

limx→0(xln(x))+limx→01

The second limit belongs to a constant function, hence it will always give 1 for every value of x.

But the first limit looks more complex than the second one, since it not only has a ln(x) as denominator, but also ln(0)=UNDEFINED. So, in order to find a legit solution, we use a method named L'Hôpital's rule, where you take derivatives of both nominator and denominator one by one and then continue to find the limits.

limx→0(xln(x))

L’Hopital’s rule,

limx→011x

limx→0x=0

We also had a constant limit of value 1, as when we add two of these limits,

1+0=1