Saturday, 3 September 2016

Limit limxtoinftyxtan1(f(x)/(x+g(x)))

I am investigating the limit




limxxtan1(f(x)x+g(x))



given that f(x)0 and g(x)0 as x. My initial guess is the limit exists since the decline rate of tan1 will compensate the linearly increasing x. But I'm not sure if the limit can be non zero. My second guess is the limit will always zero but I can't prove it. Thank you.



EDIT 1: this problem ca be reduced into proving that limxxtan1(M/x)=M for any MR. Which I cannot prove it yet.



EDIT 2: indeed limxxtan1(M/x)=M for any MR. Observe that



limxxtan1(M/x)=limx0tan1(Mx)x.

By using L'Hopital's rule, the right hand side gives M. So the limit which is being investigated is equal to zero for any f(x) and g(x) as long as f(x)0 and g(x)0 as x. The problem is solved.

No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find limh0sin(ha)h without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...