Question

r3-3, if xs2(x2 +1, ¡f x > 2

Answer 1

The given function is,

f( x )={ x 3 −3,x≤2 x 2 +1,x>2

Consider k be any real number, then the cases will be k<2, k=2 or k>2.

When k<2, then the function becomes,

f( k )= k 3 −3

The limit of the function is,

lim x→k f( x )= lim x→k ( x 3 −3 ) = k 3 −3

It can be observed that, lim x→k f( x )=f( k ).

Therefore, the function is continuous for all real numbers less than 2

When k=2, the function is,

f( 2 )= 2 3 −3 =5

The left hand limit of the function is,

LHL= lim x→ 2 − f( x ) = lim x→ 2 − ( x 3 −3 ) =8−3 =5

The right hand limit of the function is,

RHL= lim x→ 2 + f( x ) = lim x→ 2 + ( x 2 +1 ) =4+1 =5

It can be observed that, LHL=RHL.

Therefore, function is continuous at x=2.

When k>2, the function becomes,

f( k )= k 2 +1

The limit of the function is,

lim x→k f( x )= lim x→k ( x 2 +1 ) = k 2 +1

It can be observed that, lim x→k f( x )=f( k ).

The function is continuous for all real numbers greater than 2. Therefore, there is no point of discontinuity.