1.

Which of the following function is not continuous at x=0?

Answer»

`F(x)(1+2x)^(1//x),x NE 0`
`=e^(2),x=0`
`f(x)=sinx-cosx,xne0`
`=-1,x=0`
`f(x)=(e^(1//x)-1)/(e^(1//x)+1),x ne0`
`=-1,x=0`
`f(x)=(e^(5x)-e^(2x))/(sin3x),xne0`
`=1,x=0`

Solution :Option (a), here, `f(0)=e^(2)`
and `underset(x to 0)(lim)f(x)=underset(x to 0)(lim)(1+2x)^(1//x)=e^(2)""[becauseunderset(xto0)(lim)(1+lamdax)^(1//x)=e^(lamda)]`
`thereforef(0)=underset(xto0)(lim)f(x)=e^(2)`
`impliesf(x)` is CONTINOUS at x=0.
Option (b), here, f(0)=-1.
and `underset(xto0)(lim)FF(x)=underset(xto0)(lim)(sinx-cosx)=sin0-cos0=-1`
`thereforef(0)=underset(xto0)(lim)f(x)=-1`
`impliesf(x)` is continuous at x=0 option (c).
here `f(0)=-1`
and `underset(x to 0)(lim)f(x)=underset(x to 0)(lim)(e^(1//x)-1)/(e^(1//x)+1)=underset(xto0)(lim)(1-e^(-1//x))/(e^(-1//x))=1`
`therefore f(0) ne underset(x to 0)(lim)f(x)impliesf(x)` is not continuous option (d), here f(0)=1
and `underset(xto0)(lim)f(x)=underset(x to 0)(lim)(e^(5x)-e^(2x))/(sin3x)xx(3x)/(3x)`
`=underset(xto0)(lim)(e^(5x)-e^(2x))/(3x)xxunderset(xto0)(lim)(3x)/(sin3x)`
`(1+(5x)/(1!)+((5x)^(2))/(2!)+ . . .)-`
`=underset(xto0)(lim)((1+(2x)/(1!)+((2x)^(2))/(2!)+ . .))/(3x)""(because underset(xto0)(lim)(3x)/(sin3x)=1)`
`=underset(x to 0)(lim)(((3x)/(1!)+(21x^(2))/(2!)+ . . .))/(3x)=1`
`thereforef(0)=underset(xto0)(lim)f(x)=1`.
`impliesf(x)` is continuous at x=0.


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