SOLUTION: Find the limit as x approaches pi/4 of (sin(x)-cos(x))/(cos(2x)). I am told that my answer should be a numeric value. Any help is greatly appreciated!! Best, Juicy

Algebra ->  Trigonometry-basics -> SOLUTION: Find the limit as x approaches pi/4 of (sin(x)-cos(x))/(cos(2x)). I am told that my answer should be a numeric value. Any help is greatly appreciated!! Best, Juicy      Log On


   

Question 1057730: Find the limit as x approaches pi/4 of (sin(x)-cos(x))/(cos(2x)).
I am told that my answer should be a numeric value.
Any help is greatly appreciated!!
Best,
Juicy
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!

The limit is indeterminate (0/0) so I would use L'Hospital's Rule.
L'Hospital's Rule says that Lim(x->a) { f(x)/g(x) } = Lim(x->a) {f'(x)} / Lim(x->a){g'(x)}
Yes, it means you take the derivative of the top and bottom and figure out those limits individually, then you can divide (very powerful). It only works for 0/0, infinity/infinity and a few select indeterminate forms.
d(sin(x))
——— = cos(x)
dx
d(cos(x))
——— = -sin(x)
dx
d(cos(2x))
———— = -2sin(2x)
dx

So the original limit = Lim { cos(x) + sin(x) } / Lim { -2 sin(2x) }
x->pi/4 x->pi/4
= sqrt(2) / (-2)
= -0.707106