Evaluate $\lim\limits_{x \to 3^-} f(x)$ and $\lim\limits_{x \to 3^+} f(x)$ for the function: $f(x) = \frac{|x - 3|}{x - 3}$

$(−1, 1)$

The correct answer is Option (1) → $(−1, 1)$ ##

Left-hand limit $\lim\limits_{x \to 3^-} f(x)$:

For $x < 3$, $x - 3$ is negative, so $|x - 3| = -(x - 3)$.

Thus, for $x < 3$:

$f(x) = \frac{-(x - 3)}{x - 3} = -1$

So, as $x \to 3^-$, $f(x) = -1$.

$\lim\limits_{x \to 3^-} f(x) = -1$

Right-hand limit $\lim\limits_{x \to 3^+} f(x)$:

For $x > 3$, $x - 3$ is positive, so $|x - 3| = x - 3$.

Thus, for $x > 3$:

$f(x) = \frac{x - 3}{x - 3} = 1$

So, as $x \to 3^+$, $f(x) = 1$.

$\lim\limits_{x \to 3^+} f(x) = 1$