Help solving lim x--->pi/4 (1 - tan(x))/(sin(x) - cos(x)) ?

Question

I need help with this problem. No matter how I try to simplify I still seem to end up with a zero on the bottom. It has been 15 years since I took algebra and my trig class was a 5 week class this summer so please be gentle and explain your steps.

Answer 1

My best advice for these types of problems is to rewrite all the standard trig functions in terms of sine and cosine.  In this case,

tan(x) = sin(x) / cos(x).

The numerator can then be written as:

1 - tan(x) = 1 - sin(x) / cos(x)

= 1/cos(x) * (cos(x) - sin(x))

= -1/cos(x) * (sin(x) - cos(x))

Note that when we divide by the denominator, we get:

-1/cos(x) * (sin(x) - cos(x)) / (sin(x) - cos(x)) = -1/cos(x).

Evaluating the limit as x->pi/4 gives:

-1 / cos(pi/4) = -1 / (1 / sqrt(2)) = -sqrt(2).