limits and continuity examples

📚 Quick Study Guide

• 🔍 Limit Definition: The limit of a function $f(x)$ as $x$ approaches $c$ is $L$, written as $\lim_{x \to c} f(x) = L$, if $f(x)$ gets arbitrarily close to $L$ as $x$ gets sufficiently close to $c$.
• 💡 Continuity Definition: A function $f(x)$ is continuous at $x = c$ if the following three conditions are met: 1) $f(c)$ is defined, 2) $\lim_{x \to c} f(x)$ exists, and 3) $\lim_{x \to c} f(x) = f(c)$.
• 📝 Limit Laws:
  • Sum/Difference: $\lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)$
  • Constant Multiple: $\lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x)$
  • Product: $\lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$
  • Quotient: $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$, provided $\lim_{x \to c} g(x) \neq 0$
• 📈 Types of Discontinuities:
  • Removable Discontinuity: A discontinuity that can be 'removed' by redefining the function at that point.
  • Jump Discontinuity: The function 'jumps' from one value to another at a certain point.
  • Infinite Discontinuity: The function approaches infinity at a certain point.
• 🧪 Intermediate Value Theorem (IVT): If $f$ is continuous on the closed interval $[a, b]$ and $k$ is any number between $f(a)$ and $f(b)$, then there exists at least one number $c$ in $(a, b)$ such that $f(c) = k$.

Practice Quiz

1. Question 1: What is the value of $\lim_{x \to 2} (x^2 + 3x - 1)$?
1. 5
2. 9
3. 10
4. 11
2. Question 2: For what value of $k$ is the function $f(x) = \begin{cases} x^2, & x \le 1 \\ kx, & x > 1 \end{cases}$ continuous at $x = 1$?
1. 0
2. 1
3. 2
4. 3
3. Question 3: What type of discontinuity does the function $f(x) = \frac{x^2 - 4}{x - 2}$ have at $x = 2$?
1. Jump Discontinuity
2. Infinite Discontinuity
3. Removable Discontinuity
4. Essential Discontinuity
4. Question 4: Find $\lim_{x \to 0} \frac{\sin(x)}{x}$.
1. 0
2. 1
3. $\infty$
4. Does Not Exist
5. Question 5: Which of the following functions is continuous everywhere?
1. $f(x) = \frac{1}{x}$
2. $f(x) = \tan(x)$
3. $f(x) = |x|$
4. $f(x) = \frac{1}{x^2 - 1}$
6. Question 6: Given $f(x) = x^3 - 2x + 1$, does there exist a value $c$ in the interval $[0, 1]$ such that $f(c) = 0$ according to the Intermediate Value Theorem?
1. Yes
2. No
3. Cannot be determined
4. Only if c = 0
7. Question 7: Evaluate $\lim_{h \to 0} \frac{(3+h)^2 - 9}{h}$.
1. 0
2. 3
3. 6
4. Does Not Exist