f(x)=12x+ 3, if x 22х-3, if x>26.
The given function is,
f ( x ) = { 2 x + 3 , x ≤ 2 2 x − 3 , x > 2
Consider k be any real number, then the possible cases will be k < 2 , k = 2 or k > 2 .
When k < 2 , the function becomes,
f ( k ) = 2 k + 3
The limit of the function is,
lim x → k f ( x ) = lim x → k ( 2 x + 3 ) = 2 k + 3
It can be observed that, lim x → k f ( x ) = f ( k ) .
Therefore, the function is continuous for all real numbers smaller than 2.
When k = 2 , the left hand limit of the function is,
LHL = lim x → 2 − f ( x ) = lim x → 2 − ( 2 x + 3 ) = 7
The right hand limit of the function at k = 2 is,
RHL = lim x → 2 + f ( x ) = lim x → 2 + ( 2 x − 3 ) = 1
It can be observed that, at x = 2 , LHL ≠ RHL .
Therefore, the function is discontinuous at x = 2 .
When k > 2 , the function becomes,
f ( k ) = 2 k − 3
The limit of the function is,
lim x → k f ( x ) = lim x → k ( 2 x − 3 ) = 2 k − 3
It can be observed that, at k > 2 , lim x → k f ( x ) = f ( k ) .
Therefore, the function is continuous for all real numbers greater than 2.
The given function is,
Consider
When
The limit of the function is,
It can be observed that,
Therefore, the function is continuous for all real numbers smaller than 2.
When
The right hand limit of the function at
It can be observed that, at
Therefore, the function is discontinuous at
When
The limit of the function is,
It can be observed that, at
Therefore, the function is continuous for all real numbers greater than 2.
