Limit as t-->0 of 2t/tan(t)?

It is easier to work with sines and cosines.
tan(t) = sin(t)/cos(t)
I am not sure if you have learned the trig limit proofs yet, but lim t-->0 of t/sin(t) is always equal to 1. There is a proof for it.
And, lim t-->0 of cos(t) is 1 because cos(0) = 1.

So,
= lim t-->0 2t / [sin(t)/cos(t)]
= 2tcos(t) / sin(t)
= 2cos(t) [t/sin(t)]
= 2cos(t) * 1
= 2*1*1
= 2
And limit of a constant is always the constant, so the limit of 2 is 2!
(DON'T forget the limits before every line, or your teacher will yell at you :) )

Answer: lim t-->0 2t/tan(t) = 2

phew, ALL DONE!

I hope that helps!