Question

lim (2log(1+x)-log(1+2x))/x^2
x tends to 0.
Find limiting value..Please explain

Answer 1

Answer:

As you note this is a "0×−∞", which is indeterminate, so we can use L'Hopital's Rule. But first, we should follow Babak S' suggestion, observing that

xlogx=logx1/x.

Taking the limit, we obtain

limx→0xlogx=limx→0logx1/x=LHlimx→01/x−1/x2=limx→0−x2x=limx→0−x=0.

If you need to brush up on L'Hopital's Rule, you may want to consider watching Adrian Banner's lecture on the topic