Limits of tan?

Question

Can any one explain when x tends to zero what the limit of

tanx
------
x

Do I need to split to tan into sin/cos ?

Answer 1

No , It's a formula , that when x tends to zero , lim tanx /x , is one . Like the standard result of , when x tends to zero , lim sin x/ x , is one . It's a standard result . And if you want the proof , then ....it's a tough one . I can give you the proof . But the answer is one , as it is a formula .

Answer 2

Not at all! the proof is also easy... just draw a right triangle with a small angle x. call s as the side against angle x, and the side perpe to s as r. now as the angle x becomes smaller and smaller, the triangle will tend to b come a sector of a circle, where s = r. x, x in radians, curve length being equal to radians multiplied by radius.
thus, tan x, which is s/r now tends to equal x,
therefore lim (tanx)/x = 1
x--0
ofcourse we assumed x is small so we introduced the limits symbol...

Answer 3

tan(x) = sin(x)/cos(x) [or in/co :-)]

cos(x) is a continuous function, so lim x->0 cos(x) = cos(0)=1

lim x->0 sin(x)/(x * cos(x)) = lim x->0 sin(x)/x = 1