Test Questions for Intuitive Understanding of Limits

📚 Quick Study Guide

• 🔍 Definition: A limit is the value that a function approaches as the input approaches some value. We write this as $\lim_{x \to a} f(x) = L$.
• 💡 Limit Laws: These are rules that allow us to break down complex limits into simpler ones, like the sum rule, product rule, and quotient rule. For instance, $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$.
• 📝 Indeterminate Forms: These occur when direct substitution results in expressions like $\frac{0}{0}$ or $\frac{\infty}{\infty}$. L'Hôpital's Rule can often be used to evaluate these.
• ➗ L'Hôpital's Rule: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$, provided the limit on the right exists.
• 📈 One-Sided Limits: The limit from the left is denoted as $\lim_{x \to a^-} f(x)$, and the limit from the right is denoted as $\lim_{x \to a^+} f(x)$. For the limit to exist at $x=a$, both one-sided limits must exist and be equal.
• 🛑 Limits at Infinity: We look at what happens to $f(x)$ as $x$ becomes very large (positive or negative). For example, $\lim_{x \to \infty} \frac{1}{x} = 0$.

🧪 Practice Quiz

1. What is the value of $\lim_{x \to 2} (x^2 + 3x - 1)$?
   1. 5
   2. 9
   3. 10
   4. 11
2. What is the value of $\lim_{x \to 0} \frac{\sin(x)}{x}$?
   1. 0
   2. 1
   3. $\infty$
   4. Does not exist
3. What is the value of $\lim_{x \to \infty} \frac{3x^2 + 2x + 1}{x^2 - x + 5}$?
   1. 1
   2. 2
   3. 3
   4. $\infty$
4. What is the value of $\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$?
   1. 0
   2. 1
   3. 2
   4. Does not exist
5. What is the value of $\lim_{x \to 0} \frac{e^x - 1}{x}$?
   1. 0
   2. 1
   3. $\infty$
   4. Does not exist
6. What is the value of $\lim_{x \to \infty} e^{-x}$?
   1. 0
   2. 1
   3. $\infty$
   4. -$\infty$
7. What is the value of $\lim_{x \to 2} f(x)$, where $f(x) = \begin{cases} x+1, & x < 2 \\ 4, & x = 2 \\ x^2-1, & x > 2 \end{cases}$?
   1. 1
   2. 3
   3. 4
   4. Does not exist