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Find the value of k, if the function f g...

Find the value of k, if the function f given by :
`{:(f(x)=(1-tanx)/(1-sqrt2sinx)",", "for" x nepi/4),(=k/2",","for"x=pi/4):}`
is continous at `x =pi/4*`

Text Solution

Verified by Experts

`f((pi)/(4))=k/2" "..."Given"...(1)`
`underset(xto pi//4)lim f(x)=underset(x to pi//4)lim (1-tan x)/(1-sqrt2sin x)`
`=underset(x to pi//4)lim(1-tan x)/(1`-sqrt2sinx)xx(1+sqrt2sin x)/(1+sqrt2sin x )`
`=underset(x to pi//4)lim((1-tanx)(1+sqrt2sin x))/(1-2sin^(2)x)`
`=underset(x to pi//4)lim([1-(sinx//cosx)](1+sqrt2sin x))/(sin^(2)x+cos^(2)x-2sin^(2)x)`
`=underset(x to pi//4)lim((cosx-sinx)(1+sqrt2sinx))/(cos x(cos^(2)x-sin^(2)x))`
`=underset(x to pi//4)lim((cosx-sinx)(1+sqrt2sinx))/(cos x (cosx-sinx)(cosx+sinx))`
`=underset(x to pi//4)lim(1+sqrt2sinx)/(cosx (cosx +sinx))`
`...[xtopi//4, x nepi//4therefore cos -sin xnex]`
`=(underset(xto pi//4)lim(1+sqrt2sinx))/(underset(x to pi//4)limcos x xxunderset(x to pi//4)lim (cosx+sinx))`

`=(1+sqrt2sin =(pi//4))/(cos (pi//4)[cos(pi//4)+sin(pi//4)])`
`=(1+sqrt2xx(1)/(sqrt2))/((1)/(sqrt2)((1)/(sqrt2)+(1)/(sqrt2)))=(1+1)/((1)/(sqrt2)xx(2)/(sqrt2))=2." "...(2)`
Since f is continous at `c=pi/4,`
`f((pi)/(4))=underset(x to pi//4)limf(x)`
`thereforek /2=2" "...[By(1) and(2)]`
`thereforek=4.`
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