Understanding DNE Limits: Illustrated Examples for Calculus

📚 Quick Study Guide

• 🔍 A limit exists at a point if and only if the left-hand limit and the right-hand limit at that point are equal.
• 💡 If the left-hand limit and the right-hand limit are not equal, or if either limit approaches infinity, the limit Does Not Exist (DNE).
• 📝 Common scenarios where limits DNE include: jump discontinuities, vertical asymptotes, and oscillating functions.
• 📈 Jump Discontinuity: The function 'jumps' from one value to another at a specific point.
• 🧪 Vertical Asymptote: The function approaches infinity (or negative infinity) as x approaches a certain value.
• 🎢 Oscillating Function: The function oscillates infinitely between two values, never settling on a single limit.
• 🔢 Notation: Left-hand limit: $\lim_{x \to a^-} f(x)$, Right-hand limit: $\lim_{x \to a^+} f(x)$. For the limit to exist: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$

🤔 Practice Quiz

1. Question 1: For what type of discontinuity does the limit most commonly DNE?
1. A. Removable Discontinuity
2. B. Point Discontinuity
3. C. Jump Discontinuity
4. D. Continuous Function


2. Question 2: What is the limit of $f(x) = \frac{|x|}{x}$ as $x$ approaches 0?
1. A. 1
2. B. -1
3. C. 0
4. D. DNE


3. Question 3: Consider the function $f(x) = \begin{cases} 1, & x > 0 \\ -1, & x < 0 \\ \end{cases}$. What is $\lim_{x \to 0} f(x)$?
1. A. 1
2. B. -1
3. C. 0
4. D. DNE


4. Question 4: Which of the following functions has a limit that DNE as $x$ approaches 0?
1. A. $f(x) = x^2$
2. B. $f(x) = \sin(x)$
3. C. $f(x) = \frac{1}{x}$
4. D. $f(x) = \cos(x)$


5. Question 5: What happens to the limit when a function has a vertical asymptote at $x = a$?
1. A. The limit always exists and is equal to 0.
2. B. The limit always exists and is equal to infinity.
3. C. The limit exists if the function is continuous at $x = a$.
4. D. The limit typically DNE.


6. Question 6: Evaluate $\lim_{x \to 0} \sin(\frac{1}{x})$.
1. A. 0
2. B. 1
3. C. -1
4. D. DNE


7. Question 7: For the piecewise function $g(x) = \begin{cases} x+2, & x \leq 1 \\ x^2, & x > 1 \\ \end{cases}$, does $\lim_{x \to 1} g(x)$ exist?
1. A. Yes, and it equals 1.
2. B. Yes, and it equals 3.
3. C. Yes, and it equals 2.
4. D. No, it DNE.