If the function f(x) is continuous in the interval `[-2, 2].` find the values of a and b where `{:(f(x)=(sinax)/(x)-2," ,for "-2 le x lt0),(=2x+1," ,for "0 le x le1),(=2bsqrt(x^(2)+3)-1," ,for " 1 lt x le 2):}`

Explanation

`underset(xrarr0^(-))(lim)f(x)=underset(xrarr0^(-))(lim)(sinax)/(x)-2`
`=underset(xrarr0^(-))(lim)(a(sinax)/(ax)-2)`
`=axx1-2 (because underset(xrarr0)(lim)(sinx)/(x)=1)`
`=a-2`
`underset(xrarr0^(+))(lim)f(x)=underset(xrarr0^(+))(lim)(2x+1)`
`=1`
Function is continuous at x = 0
`underset(xrarr0^(-))(lim)f(x)=underset(xrarr0^(+))(lim)f(x)`
`a-2=1`
`a = 3`
Again, `underset(xrarr1^(-))(lim)f(x)=underset(xrarr1^(-))(lim)(2x+1)`
= 3
`underset(xrarr1^(+))(lim)f(x)=underset(xrarr1^(+))(lim)2bsqrt(x^(2)+3-1)`
`=2b sqrt4-1=4b-1`
Function is continuous at x = 1
Then `4b-1=3`
`rArr" "4b=4`
`rArr" "b=1.`