Why does changing the operator in $\lim_{h\to0}$ alter the result of this function? Let $f(x) = |x|$.
Attempting to differentiate $f(x)$ at 0 will fail because the limit does not exist at 0 as the left and right side are unequal.
$$\lim_{h\to0}\dfrac{f(0+h)-f(0)}{h}$$
However, my problem is about understanding why we get a different answer on the left side and the right side.
Left side :
$$\lim_{h\to0+}\frac{f(0+h)-f(0)}{h} = \lim_{h\to0+}\frac{h-0}{h} = 1$$
Right Side :
$$\lim_{h\to0-}\frac{f(0+h)-f(0)}{h} = \lim_{h\to0-}\frac{-h-0}{h} = -1$$
Why does changing the operator in $\lim_{h\to0}$ change the result of these equations? More specifically, why do I have to add $-$ before $h$ on the right side?

For the right side limit $h \gt 0,$ so $|0+h|=h$.  For the left side limit, $h \lt 0,$ so $|0+h|=-h$.  It is just the result of applying the absolute value.  I think we are prone to intuitively think variables are positive, but that is not the case for the left side limit.
