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):}`

If the function f(x) is continuous in the interval `[-2, 2].` find the

1.	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):}`
Answer» `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.`