Can someone please explain how you find the limit of xlnx as x approaches 0? Thanks!!?

Question

limit xlnx = ?
x-->0

Answer 1

you can write:
xlnx = lnx / (1/x)
in this way you can use L'Hopital's rule,
and take the derivative of the numerator and divide it by de derivative of the denumerator:

lim x lnx = lim lnx / (1/x) = lim 1/x / (-1/x^2) = lim -x =0
x-->0

Answer 2

this is of the from 0 * inf

lim x->0 x ln x
= x->0 ln x/ (1/x)

this is of the form inf/inf
differentiate both num and denom
(1/x) /(-1/x^2) = -x =0

Answer 3

the value of natural log zero is undefined, but lnX approaches zero as x approaches zero. Furthermore, you are multiplying the natural log of a small number by that same small number, hastening its approach to zero; but the limit is still zero all the same