f(x)-Ikx +1,ไม-5if5if x > 529,at 5at X-,
The expression for the function f is defined as,
f ( x ) = { k x + 1 , if x ≤ 5 3 x − 5 , if x > 5 }
The left hand limit of the function at x = 5 is,
lim x → 5 − f ( x ) = lim x → 5 − ( k x + 1 ) = lim x → 5 − h ( k x + 1 ) = lim h → 0 ( k ( 5 − h ) + 1 ) = 5 k + 1
The right hand limit of the function at x = 5 is,
lim x → 5 + f ( x ) = lim x → 5 + ( 3 x − 5 ) = lim x → 5 + h ( 3 x − 5 ) = lim h → 0 ( 3 ( 5 + h ) − 5 ) = lim h → 0 ( 15 + 3 h − 5 )
Solve for the right hand limit.
lim x → 5 + f ( x ) = 15 − 5 = 10
The exact value of the function for x = 5 is,
f ( x = 5 ) = k x + 1 = 5 k + 1
The condition for continuity of the function f at x = 5 is fulfilled if left hand limit, right hand limit and the value at that specified point are equal.
It is observed that,
lim x → 5 − f ( x ) = lim x → 5 + f ( x ) = f ( x = 5 ) 5 k + 1 = 10 = 5 k + 1 5 k = 9 k = 9 5
The expression for the function
The left hand limit of the function at
The right hand limit of the function at
Solve for the right hand limit.
The exact value of the function for
The condition for continuity of the function
It is observed that,
