Let f(x) = {(x - |x|)/x, x ≠ 0, 2, x = 0, Show that lim(x→0) does not exist.

Question

Let f(x) = \(\begin{cases} \cfrac{\text x-|\text x|}{\text x},\quad \text x\neq0\\ 2,\quad\text x=0 \end{cases} \)

{(x - |x|)/x, x ≠ 0, 2, x = 0,

Show that \(\lim\limits_{\text x \to0} \) does not exist.

Answer 1

f(x) = \(\begin{cases} \cfrac{\text x-|\text x|}{\text x},\quad \text x\neq0\\ 2,\quad\text x=0 \end{cases} \)

{(x - |x|)/x, x ≠ 0, 2, x = 0,

Left Hand Limit(L.H.L.):

Right Hand Limit(R.H.L.):

Thus, \(\lim\limits_{\text x \to0} \) f(x) does not exist.