Question

evaluate limit x-> 0 (tan 2x- sin 2x)/ x^3

Answer 1

Lim(x→ 0) { tan2x - sin2x }/x³

check which form of limit apply here,
put x = 0
0/0 hence, we can apply L- HOSPITAL rule ,

Lim(x→0) {tan2x - sin2x}/x³
differentiate separately numerator and denominator wrt x .

= Lim(x→0) { 2sec²2x - 2cos2x}/3x²

= 2/3× Lim(x→0) { 1 - cos³2x }/cos²2x.x²

= 2/3 × Lim(x→0) {(1 -cos2x)(1 + cos²2x +cos2x}/cos²2x.x²

= 2/3 × Lim(x→0){ (2sin²x/x²)(1+cos²2x + cos2x)}/cos²2x

we know,
Lim(f(x)→0) sinf(x)/f(x) = 1

= 4/3 × Lim(x→0) {(sinx/x)² × (1+cos²2x + cos2x)/cos²2x }

= 4/3 × 1 × ( 1 + 1 + 1)/1

= 4/3 × 1 × 3

= 4 ( answer )