Question

[-2, ifxs-116. f(x)-2x, if -i

Answer 1

The given function is,

f( x )={ −2   ,  x≤−1 2x   ,  −1<x≤1 2     ,  x>1

Consider k be any real number, then the cases will be k<−1, k=−1, −1<k<1, k=1 or k>1

When k<−1, then the function becomes,

f( k )=−2

The limit of the function is,

lim x→k f( x )= lim x→k ( −2 ) =−2

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

Therefore, the function is continuous for all real number less than −1.

When k=−1, then the function becomes,

f( −1 )=−2

The left hand limit of the function is,

LHL= lim x→ 1 − f( x ) = lim x→ 1 − ( −2 ) =−2

The right hand limit of the function is,

RHL= lim x→ 1 + f( x ) = lim x→ 1 + ( 2x ) =−2

It can be observed that, LHL=RHL.

Therefore, the function is continuous at x=−1.

When −1<k<1, the function becomes,

f( k )=2k

The limit of the function is,

lim x→k f( x )= lim x→k ( 2x ) =2k

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

Therefore, the function is continuous at −1<k<1.

When k=1, the left hand limit of the function is,

LHL= lim x→ 1 − f( x ) = lim x→ 1 − ( 2x ) =2

The right hand limit of the function is,

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

It can be observed that, LHL=RHL.

Therefore, the function is continuous at x=1.

When k>1, the function becomes,

f( k )=2

The limit of the function is,

lim x→k f( x )= lim x→k ( 2 ) =2

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

Therefore, the function is continuous for all points greater than 1.