How you find the limit of (1/x-1/|x|) where the x approaches 0?

1 Answer
Apr 12, 2018

#"The limit in Question does not exist"#.

Explanation:

Knowing that, for the function #f(x)=1/x-1/|x|#,

#lim_(x to 0)f(x)" exists "iff lim_(x to 0-)f(x)=lim_(x to 0+)f(x)...(lambda)#.

As regards #lim_(x to 0+)f(x)#, let us note that,

#"As "x to 0+, x gt 0, :. |x|=x, :. f(x)=1/x-1/x=0#

#:. lim_(x to 0+) f(x)=0..................(lambda_1)#.

On the other hand, as #x to 0-, x lt 0, :. |x|=-x, :. f(x)=1/x-(-1/x)=2/x#.

Clearly, #lim_(x to 0-)f(x) to -oo............(lambda_2)#.

Taking into a/c of #(lambda), (lambda_1) and (lambda_2)#,

we conclude that #lim_(x to 0)f(x)" does not exist"#.