Question

What is the limit as x approaches 0 of `(1-tan(x))/(sin(x)-cos(x))`?

Answer 1

`lim_(x->0)(1-tan(x))/(sin(x)-cos(x))=-1`

You can use the rule about the limit of a quotient equal to the quotient of the limits:
`lim_(x->a)f(x)/g(x)=(lim_(x->a)f(x))/(lim_(x->a)g(x))`
(with `lim_(x->a)g(x)` not equal to zero)
and substitute directly for `x=0` with:

`tan(0)=0`
`sin(0)=0`
`cos(0)=1`

Graphically:

This function can also be simplified by using the fact that: `tan(x)=sin(x)/cos(x)`
So that:
`(1-tan(x))/(sin(x)-cos(x))=(1-sin(x)/cos(x))/(sin(x)-cos(x)=`

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