One-Sided Limits — Question 4

Question 4

Let the function $f(x)$ be defined as: $$f(x) = \begin{cases} x^2 - 4, & x < 1 \\ 2x + 1, & x \geq 1 \end{cases}$$

(a) Find $\lim_{x \to 1^-} f(x)$

(b) Find $\lim_{x \to 1^+} f(x)$

(c) Does $\lim_{x \to 1} f(x)$ exist? Explain.

Question 4 - Solution

We are given: $$f(x) = \begin{cases} x^2 - 4, & x < 1 \\ 2x + 1, & x \geq 1 \end{cases}$$

(a) Left-hand limit:

For $x < 1$, we use $f(x) = x^2 - 4$: $$\lim_{x \to 1^-} f(x) = 1^2 - 4 = -3$$ $$\boxed{\lim_{x \to 1^-} f(x) = -3}$$

(b) Right-hand limit:

For $x \geq 1$, we use $f(x) = 2x + 1$: $$\lim_{x \to 1^+} f(x) = 2(1) + 1 = 3$$ $$\boxed{\lim_{x \to 1^+} f(x) = 3}$$

(c) Two-sided limit:

$$\lim_{x \to 1^-} f(x) = -3 \quad \text{and} \quad \lim_{x \to 1^+} f(x) = 3$$ Since the one-sided limits are not equal, the overall limit does not exist: $$\boxed{\lim_{x \to 1} f(x) \text{ does not exist}}$$