Lost on this Trig limit...

[eric_f]

Hi All,

I'm having issues with this limit problem:



I know ultimately I'm trying to end up with forms of the limit of sin(x)/x that can equal one, but the solution guide loses me at the first step...

I get to this point and confusion sets in:



Can someone nudge me along?


Thanks!

 

[Bob Brown MSEE]

(sin4x)(cosx)/(xcos4x)
note:
sin4x -> 4x
cosx -> 1
cos4x -> 1
x -> x

 

[eric_f]

OK, I think I got it! Can someone verify my solution process?



~Eric

 

[DrPhil]

Your first line is ok, but in the second line you have a spurious factor of 4 in the numerator, then 2 spurious factors of 4 in the denominator.

That 4 carries through in the denominator until the last line - where it just as suddenly vanishes again. The expression before you take the limit should be

\(\displaystyle \displaystyle \lim_{t\to 0} \left( \dfrac{\sin{4t}}{t} \times \dfrac{\cos{t}}{\cos{4t}}\right) \)

This follows from the original statement of the problem simply by replacing tan = sin/cos and sec = 1/cos.