Question

find the limiting value of Lim x tends to 0 (x.lnx)​

Answer 1

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