Self-assessment: Does a limit exist? Pre-Calculus quiz

📚 Quick Study Guide

• ♾️ A limit describes the value that a function approaches as the input (e.g., $x$) approaches some value.
• 📈 Graphical Interpretation: Examine the function's graph near the point of interest. Does the function approach a specific $y$-value from both sides?
• ➕ One-Sided Limits: $\lim_{x \to a^-} f(x)$ is the limit as $x$ approaches $a$ from the left. $\lim_{x \to a^+} f(x)$ is the limit as $x$ approaches $a$ from the right. For a limit to exist, both one-sided limits must exist and be equal.
• ➗ Indeterminate Forms: Expressions like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ are indeterminate. Use techniques like factoring, rationalizing, or L'Hôpital's Rule (if applicable) to evaluate the limit.
• 📐 Limit Laws: Several laws allow you to break down complex limits: e.g., $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$.
• 🧮 Direct Substitution: If $f(x)$ is continuous at $x = a$, then $\lim_{x \to a} f(x) = f(a)$. Try this first!
• 💡 When Limits Fail to Exist: The function approaches different values from the left and right, the function oscillates wildly, or the function increases or decreases without bound.

Practice Quiz

1. What is the value of $\lim_{x \to 2} (x^2 + 3x - 1)$?
   1. 5
   2. 9
   3. 1
   4. 16
2. Evaluate $\lim_{x \to 0} \frac{\sin(x)}{x}$.
   1. 0
   2. 1
   3. $\infty$
   4. -1
3. What is $\lim_{x \to \infty} \frac{3x^2 + 2x - 1}{x^2 - 5}$?
   1. $\infty$
   2. 0
   3. 3
   4. -5
4. Find $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$.
   1. 0
   2. 6
   3. $\infty$
   4. 3
5. Determine $\lim_{x \to 0} \frac{1}{x}$, if it exists.
   1. 0
   2. 1
   3. Does not exist
   4. $\infty$
6. Evaluate $\lim_{x \to 1} \frac{x - 1}{\sqrt{x} - 1}$.
   1. 0
   2. 1
   3. 2
   4. $\infty$
7. Find $\lim_{x \to -2} \frac{x+2}{x^2 - 4}$.
   1. 0
   2. $\frac{1}{4}$
   3. $-\frac{1}{4}$
   4. Does not exist