Question

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<=x<=0`

`=2x+1, for 0<=x<=1`

`=2bsqrt(x^2+3)-1, for 1<x<=2`

Answer 1

Since the function f (x) is continuous in the interval [-2,2]

 f is continuous at in x = 0 and x = 1
(i) continuity at x = 0

`lim_(x->0)f(x)=lim_(x->0)((sinax)/x-2)`

`=lim_(x->0)((sinax)/(ax)a-2)`

=a(1)-2

=a-2

f (x)= 2x +1, for 0<= x <=1 ...(i)
f(0)=2(0)+1=1

f is continuous at x=0

`lim_(x->0^-)f(x)=f(0)`

a-2=1

a=3

(ii) Continuity at x = 1

From (i), f(1)=3

`lim_(x->1)f(x)=lim_(x->1^+)(2bsqrt(x^2+3)-1)`

`=2blim_(x->1)sqrt(x^2+3)-1`

`=2bsqrt(1+3)-1=4b-1`

f is continuous at x = 1

`lim_(x->1)f(x)=f(1)`

4b-1=3

4b=4

b=1