Question

suppose the function is defined by
f(x)= (|x-3|) / x-3 , x not equal 3 , then the left hand
0, x=3
then the limit of f(x) at x=3 is

Answer 1

Given :  f(x)  = (| x - 3| /( x - 3)  , x ≠ 3

f(x) = 0   at x = 3

To Find : Limit of f(x)  at x = 3

Solution:

 f(x)  = (| x - 3| /( x - 3)

for LHL  x = 3 - a  where a is + ve a → 0

for RHL  x = 3 + a  where a is + ve a → 0

| x |  = x  if x ≥ 0

       -x  if x < 0

LHL  

Lim a → 0  (| 3 - a - 3|)/(3 - a - 3)

= Lim a → 0  (| -a|)/(  - a )

= Lim a → 0  a/(-a)

=   - 1

RHL  

Lim a → 0  (| 3 + a - 3|)/(3 + a - 3)

= Lim a → 0  (| a|)/(  a )

= Lim a → 0  a/( a)

=  1

LHL ≠ RHL

Hence limit of f(x) at x=3  does not exist  , function is not continuous at  x = 3

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