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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)=(8^(x)-2^(x))/(k^(x)-1)",","for"x ne0),(=2",","for"x=0):}`
is continous at `x=0.`

Text Solution

Verified by Experts

`f(0)=2" "..."Given"...(1)`
`underset(x to 0)limf(x)=underset(x to 0)lim(8^(x)-2^(x))/(k^(x)-1)`
`=underset(xto0)lim(2^(x)*4^(x)-2^(x))/(k^(x)-1)`
`=underset(xto0)lim(2^(x)(4^(x)-1))/(k^(x)-1)`
`=underset(x to 0)lim(2^(x)((4^(x)-1)/(x)))/(((k^(x)-1)/(x)))" "...[x to0, x ne0]`
`=(underset(xto0)lim2^(x)xxunderset(xto0)lim (4^(x)-1)/(x))/(underset(xto0)lim(k^(x)-1)/(x))`
`=(2^(0)xxlog4)/(logk)" "...[because underset(xto0)lim(x^(x)-1)/(x)=loga]`
Since f is continous at `x=0.`
`underset(x to 0) lim f(x)=f(0)`
`therefore(log4)/(logk)=2`
`thereforelogk=1/2log4=log2`
` thereforek=2.`
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