Question

Find the indicated limit.

lim tan(x) / sin (2x)
x->0

Seemed like a relatively simple problem, which I broke up to lim tan(x) / lim (sin2x), which I assumed would then be 0/0. But, when plugging it into my TI-89 I get the answer of 1/2.

Please, help!

Answer 1

1/2

Answer 2

first multiply and divide by x

Answer 3

it comes to

tan(x)/ x * x/sin(2x)

tanx/ x approaches 1, x/sin2x approaches 1/2

so limit is

1 * 1/2=1/2

Answer 4

Thank you for the response, but why? I see what you have done, but I have no idea why you did it and no idea why my method was incorrect (only know that it is incorrect).

Answer 5

Note that tan(x)=sin(x)/cos(x) and sin(2x)=2sin(x)cos(x)

So we have sin(x)/[2sin(x)cos(x)cos(x)]

=1/2(cos(x))^2...plugging in 0 yields 1/2, which is our limit (since it's not an indeterminate form).

At least that's how I'd approach it...