Test questions on the existence of a limit in calculus

📚 Quick Study Guide

🔍 Understanding Limits: The limit of a function $f(x)$ as $x$ approaches $c$ is the value that $f(x)$ gets closer and closer to as $x$ gets closer and closer to $c$. We write this as $\lim_{x \to c} f(x) = L$. 💡 Formal Definition ($\epsilon-\delta$): For every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - c| < \delta$, then $|f(x) - L| < \epsilon$. 📝 Limit Laws: These laws allow us to compute limits of combinations of functions. Key laws include: * $\lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$ * $\lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$ * $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$ (provided $\lim_{x \to c} g(x) \neq 0$) 🧮 Indeterminate Forms: Expressions like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ are called indeterminate forms. L'Hôpital's Rule can often be applied in these cases. L'Hôpital's Rule states that if $\lim_{x \to c} f(x) = 0$ and $\lim_{x \to c} g(x) = 0$, then $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$, provided the limit on the right exists. 📈 One-Sided Limits: $\lim_{x \to c^-} f(x)$ is the left-hand limit (as $x$ approaches $c$ from values less than $c$), and $\lim_{x \to c^+} f(x)$ is the right-hand limit (as $x$ approaches $c$ from values greater than $c$). For a limit to exist, both one-sided limits must exist and be equal.

🧪 Practice Quiz

1. What is the limit of the function $f(x) = 3x^2 + 2x - 1$ as $x$ approaches 2?
   1. A) 11
   2. B) 15
   3. C) 16
   4. D) The limit does not exist
2. Find the limit of $f(x) = \frac{\sin(x)}{x}$ as $x$ approaches 0.
   1. A) 0
   2. B) 1
   3. C) $\infty$
   4. D) The limit does not exist
3. Evaluate $\lim_{x \to \infty} \frac{2x^2 + 3x - 1}{x^2 - 5x + 4}$.
   1. A) 0
   2. B) 1
   3. C) 2
   4. D) $\infty$
4. Determine $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$.
   1. A) 0
   2. B) 3
   3. C) 6
   4. D) The limit does not exist
5. Find the limit of $f(x) = \frac{x+2}{x-2}$ as $x$ approaches 2 from the right (i.e., $\lim_{x \to 2^+} \frac{x+2}{x-2}$).
   1. A) 0
   2. B) 1
   3. C) $\infty$
   4. D) -$\infty$
6. What is the limit of $f(x) = e^x$ as $x$ approaches $-\infty$?
   1. A) 0
   2. B) 1
   3. C) $\infty$
   4. D) -$\infty$
7. Evaluate $\lim_{h \to 0} \frac{(3+h)^2 - 9}{h}$.
   1. A) 0
   2. B) 3
   3. C) 6
   4. D) 9