Question

1. (10 points) consider

$f(x) = \

$$\begin{cases} x + 1, & \\text{if } x \\geq 0 \\\\ -x - 2, & \\text{if } x < 0 \\end{cases}$$

$
estimate $\lim\limits_{x \to 0} f(x)$.

Explanation:

Step1: Compute right-hand limit

For $x \geq 0$, $f(x)=x+1$. $\lim_{x \to 0^+} f(x) = 0 + 1 = 1$

Step2: Compute left-hand limit

For $x < 0$, $f(x)=-x-2$. $\lim_{x \to 0^-} f(x) = -0 - 2 = -2$

Step3: Compare limits

Right-hand limit ≠ left-hand limit, so limit does not exist.

Answer:

The limit does not exist.