Find the #lim_(x to pi/4) {4sqrt3-(cos(x)+sin(x))^5}/(1-sin(2x))#?

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Answer 1 · 2017-12-27T17:48:45

See below.

#(4sqrt(3)-(cos(x)+sin(x))^5)/(1-sin(2x))=#

#(4sqrt(3)-(cos(x)+sin(x))^5)*1/(1-sin(x)#

Using:

#lim_(x->a)(f(x)*h(x))=lim_(x->a)f(x)*lim_(x->a)(h(x)#

#lim_(x->pi/4)(4sqrt(3)-(cos(x)+sin(x))^5)#

Plugging in #pi/4#

#(4sqrt(3)-(cos(pi/4)+sin(pi/4))^5)#

#(4sqrt(3)-(sqrt(2)/2+sqrt(2)/2)^5)=4(sqrt(3)-sqrt(2))#

#:.#

#lim_(x->pi/4)(4sqrt(3)-(cos(x)+sin(x))^5)=4(sqrt(3)-sqrt(2))#

#lim_(x->pi/4)(1/(1-sin(2x)))=oo#

(denominator is positive as #sin(2x)->1# )

#:.#

#lim_(x->pi/4)(4sqrt(3)-(cos(x)+sin(x))^5)/(1-sin(2x))=4(sqrt(3)-sqrt(2))*oo=oo#