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

1 Answer
Gió
Dec 23, 2014

#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:

enter image source here

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)#

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