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Evaluate: (lim)(x->(3pi)/4)(1+(tanx)^(1/...

Evaluate: `(lim)_(x->(3pi)/4)(1+(tanx)^(1/3))/(1-2cos^2x)`

Text Solution

Verified by Experts

`underset(xto(3pi)/(4))lim(1+3sqrt(tan x))/(1-2cos^(2)x)`
`underset(xto(3pi)/(4))lim(1+(tanx)^(1//3))/(-cos2x).(1-(tanx)^(1//3)+(tanx)^(2//3))/(1-(tanx)^(1//3)+(tanx)^(2//3))`
`=-underset(xto(3pi)/(4))lim(1+tanx)/((1-tan^(2)x)/(1+tan^(2)x)).(1)/(1-(-1)^(1//3)+(-1)^(2//3))`
`-underset(xto(3pi)/(4))lim(1+tan^(2)x)/(1-tanx).(1)/(3)=-(1+1)/(1-(-1)).(1)/(3)=-1/3`
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