Apply limit (theta) -> 0 Tan(theta)-Sin(theta)/sin(cube)(theta) when limit (theta) -> 0

Question

Tan(theta)-sin(theta)/sin(cube)(theta) When limit theta tends to zero.

Answer 1

Writing t for theta, tan(t)=sin(t)sec(t),

so if t≠0 we can write: (tan(t)-sin(t))/(sin(t))^3=(sec(t)-1)/(sin(t))^2.

The expansion of cos(t)=1-t^2/2 when t is small, and sin(t)=t.

Also, sec(t)=1/cos(t)=(1-t^2/2)^-1=1+t^2/2 when t is small. Therefore sec(t)-1=t^2/2 And (sin(t))^2=t^2.

So the quotient approaches t^2/2t^2=1/2 as t approaches zero.

The limit is therefore 1/2.