1.

If the function `f(x)`defined as `f(x)`defined as `f(x)={3,x=0(1+(a x+b x^3)/(x^2)),x >0`is continuous at `x=0,`then`a=0`b. `b=e^3`c. `a=1`d. `b=(log)_e3`A. a = 0B. `b=e^(3)`C. `a=1`D. `b=log_(e)3`

Answer» Correct Answer - A::D
`underset(hrarr0)(lim)f(0+h)=underset(hrarr0)(lim)(1+(ah+bh^(3))/(h^(2)))^(1//h)`
`" "=underset(hrarr0)(lim)e^((1)/(h)ln(1+(ah+bh^(3))/(h^(2))))`
For limit to exist, we must have
`underset(hrarr0)(lim)(ah+bh^(3))/(h^(2))=0`
`rArr" "underset(hrarr0)(lim)(a+bh^(2))/(h)=0`
`therefore" "a=0`
So, we have
`underset(hrarr0)(lim)f(0+h)=underset(hrarr0)(lim)(1+bh)^(1//h)`
`" "=underset(hrarr0)(lim)(1+bh)^((1//bh)b)=e^(b)`
For f(x) to be continuous at x = 0, we must have
`underset(xrarr0^(+))(lim)f(x)=f(0)`
`rArr" "e^(b)=3`
`therefore" "b=log_(e)3`


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