Question

Evaluate the limit? `lim_(x rarr 0) (1-cos2x+tan^2x)/(xsin2x)`

Answer 1

I assume that we seek:

`L = lim_(x rarr 0) (1-cos2x+tan^2x)/(xsin2x)`

This is of an indeterminate form `0/0` so we can apply L'Hôpital's rule:

`L = lim_(x rarr 0) (d/dx(1-cos2x+tan^2x))/(d/dxxsin2x)`

`\ \ = lim_(x rarr 0) ( 0+2sin2x+2tanxsec^2x )/(2xcos2x+sin2x)`

`\ \ = 2lim_(x rarr 0) ( sin2x+tanxsec^2x )/(2xcos2x+sin2x)`

Again, this is of an indeterminate form `0/0` so we can apply L'Hôpital's rule a second time:

`L = 2lim_(x rarr 0) ( d/dx(sin2x+tanxsec^2x) )/( d/dx( 2xcos2x+sin2x) )`

`\ \ = 2lim_(x rarr 0) ( 2cos2x + 2sec^2x tan^2 x +sec^4x )/( 2cos2x - 2xsin2x+ 2cos2x )`

`\ \ = 2 ( 2 + 0 + 1 )/( 2 - 0 + 2 )`

`\ \ = 2 ( 3 )/( 4 )`

`\ \ = 3/2`