How can I solve lim t-> 0 t^3/tan^3(2t) without using l'Hôpital's rule?

[nameVoid]

lim t-> 0 ,t^3/tan^3(2t) , not seeing nay identiites to solve with, escpeted to solve not using l hospitols

 

[arildno]

Well, you might try to rewrite this as:
$$\lim_{t\to{0}}\frac{t^{3}}{\tan^{3}(2t)}=\lim_{t\to{0}}(\frac{t}{\sin(2t)})^{3}\cos^{3}(2t))=\lim_{t\to{0}}(\frac{1}{2})^{3}(\frac{2t}{\sin(2t)})^{3}\cos^{3}(2t))$$

 

[nameVoid]

still not seeing it

 

[Gib Z]

Really? What are the limits of those terms individually?

 

[nameVoid]

term containing sin(2t) 0/0 foo

 

[Gib Z]

You've never learned $$\lim_{x\to 0} \frac{\sin x}{x} = 1$$ ?

 

[nameVoid]

right, thanks