Question

How do you find the limit of `(1-tanx)/(sinx-cosx)` as `x->pi/4`?

Answer 1

We will have to simplify the function from it's current form using identities, since if we input `x = pi/4` directly, we will get a denominator of `0`. The simplification will depend on the identity `tantheta = sintheta/costheta`

`=lim_(x -> pi/4) ((1 - sinx/cosx)/(sinx - cosx))`

`=lim_(x ->pi/4) ((cosx - sinx)/cosx)/(sinx - cosx)`

`=lim_(x->pi/4) (cosx - sinx)/cosx xx 1/(sinx - cosx)`

`=lim_(x-> pi/4) (-(sinx - cosx))/cosx xx 1/(sinx - cosx)`

`=lim_(x->pi/4) -1/cosx`

`=-1/cos(pi/4)`

`=-1/(1/sqrt(2))`

`=-sqrt(2)`

Hopefully this helps!