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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)=[tan((pi)/(4)+x)^(1/x)]",", "for"x ne0),(=k",", "for" x=0):}`
is continous at `x=0.`

Text Solution

Verified by Experts

`f(0)=k" "..."Given"...(1)`
`underset(xto0)limf(x)underset(xto0)lim[tan((pi)/(4)+x)]^(1/x)`
`underset(xto0)lim((tan""(pi)/(4)+4)/(1-tan""pi/4*tanx))^(1/x)`
`=underset(xto0)lim((1+tanx)/(1-tanx))^(1/x)" "...[becausetan""pi/4=1]`
`=underset(xto0)lim[({(1+tanx)^(1/(tanx))}^(tanx/x))/({(1-tanx)^(-1/tanx)}^(-tanx/x))]`
`=({underset(xto0)lim(1+tanx)^(1/tanx)}^(underset(xto0)lim(tanx)/(x)))/({underset(xto0)lim(1-tanx)^(-1/tanx)}^(-underset(xto0)limtan/x))`
`=(e^(1))/(e^(-1))...[xto0,pmtanxto0and underset(yto0)lim(1+y)^(1/y)=e]`
`=e^(2)" "...(2)`
Since f is continous at `x=0.`
`f(0) =underset(xto0)lim f(x)`
`therefore k=e^(2) " "...[By(1) and (2)]`
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