Question

30. If f(x)-0,x0-1, x>(0For what value (s) of a does lim fo) exists?

Answer 1

Let the given function defined over a range as

f( x )={ | x |+1,x<0 0,x=0 | x |−1,x>0

We need to consider three cases: a<0 , a=0 and a>0

Case 1: when a<0

lim x→ a − f( x )= lim x→ a − ( | x |+1 ) = lim x→a ( −x+1 ) =−a+1 =−a+1 ( x<a<0 ; | x |=−x )

lim x→ a + f( x )= lim x→ a + ( | x |+1 ) = lim x→a ( −x+1 ) =−a+1 =−a+1 ( a<x<0 ; | x |=−x )

Since, lim x→ a − f( x )= lim x→ a + f( x )=( −a+1 ) .

Therefore, the limit of f( x ) exists at x=a , when a<0

Case2: when a=0

lim x→ 0 − f( x )= lim x→ 0 − ( | x |+1 ) = lim x→0 ( −x+1 ) =0+1 =1 (If x<0 ; | x |=−x )

lim x→ 0 + f( x )= lim x→ 0 + ( | x |−1 ) = lim x→0 ( x−1 ) =( 0−1 ) =−1 (If x>0 ; | x |=−x )

Since, lim x→ 0 − f( x )≠ lim x→ 0 + f( x )

So, the limit of f( x ) does not exist when a=0 .

Case 3: when a>0

lim x→ a − f( x )= lim x→ a − ( | x |−1 ) = lim x→a ( x−1 ) =a−1 =a−1 ( 0<x<a ; | x |=−x )

lim x→ a + f( x )= lim x→ a + ( | x |−1 ) = lim x→a ( x−1 ) =a−1 =a−1 ( 0<a<x ; | x |=−x )

Since, lim x→ a − f( x )= lim x→ a + f( x )=( a−1 ) .

So, the limit of f( x ) exists at x=a , when a>0 .

Thus, the limit of the function f( x ) exists for all a≠0