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NCERT Class XI Mathematics - Limits and Derivatives - Solutions

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Question : 30
Total: 72
If f (x) = {
/x/+1x<0
0x=0
/x/−1x>0
. For what value(s) of a does
lim
x→a
 f (x) exists ?
Solution:  
We have f (x) = {
/x/+1x<0
0x=0
/x/−1x>0
Case (i) : When a < 0, we have
lim
x→a
 f (x) =
lim
x→a−
 |x| + 1 =
lim
x→a−
 (- x + 1) = - a + 1
Case (ii) : When a > 0, we have
lim
x→a
 f (x) =
lim
x→a+
 (x - 1) = a - 1
Case (iii) : When a = 0, we have
=
lim
x→a+
 f (x) =
lim
x→0+
 f (x) =
lim
x→0+
 (|x| - 1) = 0 - 1 = - 1
lim
x→a−
 =
lim
x→0−
 f (x) =
lim
x→0−
 (|x| + 1) = - 0 + 1 = 1
∴
lim
x→a+
 f (x) ≠
lim
x→a−
 f (0) [Since f (0) = 0]
lim
x→a
 f (x) does not exist when a = 0.
∴
lim
x→a
 f (x) exists for all a ≠ 0.
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