Evaluate $\lim\limits_{x \to 3^-} f(x)$ and $\lim\limits_{x \to 3^+} f(x)$ for the function: $f(x) = \frac{|x - 3|}{x - 3}$
Answer & explanation
Correct answer: option 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 $x \to 3^+$, $f(x) = 1$.
$\lim\limits_{x \to 3^+} f(x) = 1$