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I am currently trying to understand some limits. |x| is not differentiable because left and right limits are different.

I also learned that |x|/x=sgn(x) (signum)

But

[tex]\lim_{\substack{x\rightarrow 0}} sgn(x)[/tex]

is not defined, ergo function abs(x) is not differentiable in 0.

But isn't sgn(x) defined to be 0 if x=0, or am I misinterpreting the term "approaching".

Another example:

let

[tex]g(y) = \begin{Bmatrix}

1, & y \not= 0 \\ 0, & y=0 \end{Bmatrix} [/tex]

and f(x)=0 for all x.

[itex]\lim_{\substack{y\rightarrow 0}} g(y)=1[/itex]

[itex]\lim_{\substack{x\rightarrow 0}} f(x)=0[/itex]

but

[itex]\lim_{\substack{x\rightarrow 0}} (g \circ f) (x) =0 [/itex]

This is example where composition of functions doesn't work as they should.

I don't understand how this composition of functions, doesn't work. Can anybody explain to me how this works? ( I found the example on the internet, but not as good explanation)

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# Limit Function Problem

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