# If the function f (x) is continuous in the interval [-2, 2],find the values of a and b where - Mathematics and Statistics

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

#### Solution

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

Concept: Definition of Continuity - Continuity in Interval - Definition
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