Question

Evaluate the limit.
lim x→0 (tanx-sinx)/x³

Answer 1

That's not even a hard expression to deal with. You should know what is tangent.

You need to know tanx=sinxcosx

Then your expression can be simplified just like that:

limx→0tanxsinx=limx→0sinxcosxsinx

=limx→0sinxcosx⋅sinx

=limx→01cosx=1

(because cos0=1)

When you got a limit expression which fits to 00form, please try to simplify it first.

If you can't, then you can try to solve it with hard way like L'Hôpital's rule or squeze theorem.

okkk .....i hope its help dis

Answer 2

 lim (x → 0) (tanx - sinx) / x^3 
= lim (x → 0) sinx / x * lim (x → 0) (1 - cosx) / (x^2 *cosx) 
= 1*lim (x → 0) 1/cosx*lim (x → 0) (1-cosx) * (1+cosx) / [x^2*(1+cosx)] 
= 1 * 1 * lim (x → 0) sin^2 x / x^2 * lim (x → 0) 1 / (1 + cosx) 
= 1 * 1 * 1 * 1/2