a+bx, x<14,b-ax, x>128. Suppose f(x)=x=1and if limf (x)-f (1) what are possible values of a and b?
Let the given function defined over their range as
f ( x ) = { a + b x , x < 1 4 , x = 1 b − a x , x > 1
We need to find the possible values of a and b from the given function.
From the definition of limits, we know that:
lim x → a f ( x ) = f ( a )
On substituting the values into the expressions, we get
lim x → 1 − f ( x ) = lim x → 1 − ( a + b x ) = lim x → 1 ( a + b x ) = ( a + b ⋅ 1 ) = ( a + b ) (1)
lim x → 1 + f ( x ) = lim x → 1 + ( b − a x ) = lim x → 1 ( b − a x ) = ( b − a ⋅ 1 ) = ( b − a ) (2)
Also it is given that,
lim x → 1 f ( x ) = f ( 1 )
The value of f ( 1 ) = 4 (3)
We know that the limit of a function at a particular point only exists if its left hand limit is equal to the right hand limit. That is:
lim x → 1 − f ( x ) = lim x → 1 + f ( x ) = lim x → 1 f ( x )
From equations (1), (2) and (3), we get
a + b = 4 (4)
And
b − a = 4 (5)
On solving equations (4) and (5), we get
2 b = 8 b = 4
Putting the value of b in equation (3), we get
a = 0
Thus, the possible value of a and b is 0 and 4 respectively.
Let the given function defined over their range as
We need to find the possible values of
From the definition of limits, we know that:
On substituting the values into the expressions, we get
Also it is given that,
The value of
We know that the limit of a function at a particular point only exists if its left hand limit is equal to the right hand limit. That is:
From equations (1), (2) and (3), we get
And
On solving equations (4) and (5), we get
Putting the value of
Thus, the possible value of
and if
f(x) = f(1) what are possible values of a and b?