27. Find lmf(x)where f(x)-lxl-5
Let the given function defined over range as
f ( x ) = | x | − 5
We have to find the value of the given function at limit x → 5 .
We consider a particular point at x = 5 to find its left hand and right hand limit.
From the definition of limits, we know that:
lim x → a f ( x ) = f ( a )
On solving the expression for the value at x = 5 ,
We know that modulus function | x | = − x when x < 0 and | x | = + x when x > 0
lim x → 5 − f ( x ) = lim x → 5 − | x | − 5 = lim x → 5 ( x − 5 ) ( | x | = + x ; x > 0 )
On applying limits:
lim x → 5 ( x − 5 ) = ( 5 − 5 ) = 0 (1)
Again,
lim x → 5 + f ( x ) = lim x → 5 + | x | − 5 = lim x → 5 ( x − 5 ) ( | x | = + x ; x > 0 )
On applying limits:
lim x → 5 ( x − 5 ) = ( 5 − 5 ) = 0 (2)
From the equations 1 and 2 we found that,
lim x → a − f ( x ) = lim x → a + f ( x )
So the limit exists.
Thus, the value of the given expression lim x → 5 | x | − 5 is 0.
Let the given function defined over range as
We have to find the value of the given function at limit
We consider a particular point at
From the definition of limits, we know that:
On solving the expression for the value at
We know that modulus function
On applying limits:
Again,
On applying limits:
From the equations 1 and 2 we found that,
So the limit exists.
Thus, the value of the given expression
