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

1 Answer
Noah G
Nov 3, 2016

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!

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