Question

How do you compute the limit of `(sinx-cosx)/cos(2x)` as `x->pi/4`?

Answer 1

Multiply by `frac{sin(x) + cos(x)}{sin(x) + cos(x)}`.

`frac{sin(x) - cos(x)}{cos(2x)} * frac{sin(x) + cos(x)}{sin(x) + cos(x)} = frac{sin^2(x) - cos^2(x)}{cos(2x)(sin(x) + cos(x))}`

Use the double angle identity for cosine: `cos^2(x) - sin^2(x) -= cos(2x)`

`frac{sin^2(x) - cos^2(x)}{cos(2x)(sin(x) + cos(x))} = frac{-cos(2x)}{cos(2x)(sin(x) + cos(x))}`

Now to evaluate the limit

`lim_{x->pi/4}frac{sin(x) - cos(x)}{cos(2x)} = lim_{x->pi/4}-frac{1}{sin(x) + cos(x)}`

`= -frac{1}{sin(pi/4) + cos(pi/4)}`

`= -frac{1}{1/sqrt2 + 1/sqrt2}`

`= sqrt2/2`