Find the relationship between a and b so that the function f defined by is continuous at x = 3.

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Solution

The given function is,

f( x )={ ax+1 , x≤3 bx+3 , x>3

The left hand limit of the function is,

LHL= lim x→ 3 − f( x ) = lim x→ 3 − ( ax+1 ) =3a+1

The right hand limit of the function is,

RHL= lim x→ 3 + f( x ) = lim x→ 3 + ( bx+3 ) =3b+3

As the function is continuous at x=3, so, LHL=RHL=f( 3 ).

3a+1=3b+3 3a=3b+2 a=b+ 2 3

Therefore, the relationship between a and b is a=b+ 2 3 .

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