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Discuss the continuity of the following ...

Discuss the continuity of the following functions at the points shown against them :
`{:(f(x)=(10^(x)+7^(x)-14^(x)-5^(x))/(1-cos4x),",for"x ne0),(=10/7, "for"x=0):}}at x=0.`

Text Solution

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`f(0)=10/7" "...("Given")....(1)`
`underset(xto0)limf(x)=underset(xto0)lim(10^(x)+7^(x)-14^(x)-5^(x))/(1-cos4x)`
`=underset(xto0)lim(10^(x)-5^(x)-14^(x)+7^(x))/(1-cos4x)`
`=underset(xto0)lim(5^(x)xx2^(x)-5^(x)-7^(x)xx2^(x)+7^(x))/(2sin^(2)2x)`
`=underset(xto0)lim(5^(x)(2^(x)-1)-7^(x)(2^(x)-1))/(2sin^(2)2x)`
`=underset(xto0)lim((2^(x)-1)(5^(x)-7^(x)))/(2sin^(2)2x)`
`=underset(xto0)lim((2^(x)-1)[(5^(x)-1)-(7^(x)-1)])/(2sin^(2)2x)`
`=underset(xto0)lim(((2^(x)-1)/(x))[((5^(x)-1)/(x))-((7^(x)-1)/(x))])/(2((sin2x)/(2x))^(2)xx4)...[becausex to0,x ne0]`
`=1/8((underset(xto0)lim(2^(x)-1)/(x))[(underset(xto0)lim(5^(x)-1)/(x))-(underset(xto0)lim(7^(x)-1)/(x))])/((underset(xto0)lim(sin2x)/(2x))^(2))`
`=1/8((log2)[log5-log7])/((1)^(2))`
`...[xto0, 2x to0andunderset(thetato0)lim(sintheta)/(theta)=1, underset(xto0)lim(a^(x)-1)/(x)=loga]`
`thereforeunderset(xto0)lim f(x)=1/8(log2)*log((5)/(7))" "...(2)`
From (1) and (2),` underset(xto0)limf(x)nef(0)`
`therefore f` isdiscontinous at `x=0.`
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