Question

How am I supposed to do this?

Answer 1

The two-sided limit exists as long as the limits from either side also exist and are equal. So to compute the limit, you have to compute the one-sided limits,

$\displaystyle\lim_{x\to0^-}f(x)$

$\displaystyle\lim_{x\to0^+}f(x)$

f(x) has a different definition and thus different behavior as x approaches 0 depending on which direction x is approaching 0:

• When x approaches 0 from the left (or from below), you use the definition of f for x < 0, so

$\displaystyle\lim_{x\to0^-}f(x) = \lim_{x\to0}(-x-2) = -2$

• When x approaches 0 from the right (or above), you use the other definition, so

$\displaystyle\lim_{x\to0^+}f(x) = \lim_{x\to0} \left(\frac x2-2\right) = -2$

The limits from either side match, so the two-sided limit exists and is also -2.