Solved Problems on One-Sided Limits: Step-by-Step Examples

📚 Quick Study Guide

🔍 One-sided limits examine the behavior of a function as it approaches a specific $x$-value from either the left or the right. ➡️ The right-hand limit is denoted as $\lim_{x \to a^+} f(x)$, indicating the limit as $x$ approaches $a$ from values greater than $a$. ⬅️ The left-hand limit is denoted as $\lim_{x \to a^-} f(x)$, indicating the limit as $x$ approaches $a$ from values less than $a$. 🧩 A two-sided limit, $\lim_{x \to a} f(x)$, exists if and only if both the left-hand limit and the right-hand limit exist and are equal. 🚧 If the left-hand limit and the right-hand limit are not equal, the two-sided limit does not exist. 📈 When evaluating one-sided limits, consider the function's behavior specifically on the indicated side of the $x$-value. 💡 Piecewise functions are common in one-sided limit problems because their definitions change at specific $x$-values.

Practice Quiz

1. Question 1: What is the value of $\lim_{x \to 2^+} (x^2 - 3)$?
   1. A) 1
   2. B) -1
   3. C) 0
   4. D) Does not exist
2. Question 2: Find $\lim_{x \to 0^-} \frac{|x|}{x}$.
   1. A) 1
   2. B) -1
   3. C) 0
   4. D) Does not exist
3. Question 3: Determine $\lim_{x \to 1^+} f(x)$ where $f(x) = \begin{cases} x+1, & x < 1 \\ x^2, & x \geq 1 \end{cases}$.
   1. A) 1
   2. B) 2
   3. C) 0
   4. D) Does not exist
4. Question 4: Evaluate $\lim_{x \to 3^-} \frac{1}{x-3}$.
   1. A) $\infty$
   2. B) $-\infty$
   3. C) 0
   4. D) 1/6
5. Question 5: What is $\lim_{x \to -2^+} (x+2)^2$?
   1. A) 4
   2. B) 0
   3. C) -4
   4. D) Does not exist
6. Question 6: Find $\lim_{x \to 4^-} \sqrt{16-x^2}$.
   1. A) 0
   2. B) 4
   3. C) 8
   4. D) Does not exist
7. Question 7: Determine $\lim_{x \to 5^+} \frac{x-5}{|x-5|}$.
   1. A) 1
   2. B) -1
   3. C) 0
   4. D) Does not exist