Solved examples: Finding limits and continuity of piecewise functions

📚 Quick Study Guide

• 🔍 Piecewise Functions: These are functions defined by multiple sub-functions, each applying to a specific interval of the domain.
• 🔢 Limits: To find the limit of a piecewise function at a point where the function's definition changes, you need to check the left-hand limit and the right-hand limit. If they are equal, the limit exists and is equal to their common value.
• ➡️ Left-Hand Limit: The limit as $x$ approaches $a$ from the left, denoted as $\lim_{x \to a^-} f(x)$.
• ⬅️ Right-Hand Limit: The limit as $x$ approaches $a$ from the right, denoted as $\lim_{x \to a^+} f(x)$.
• ✅ Existence of Limit: $\lim_{x \to a} f(x)$ exists if and only if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$.
• 🔗 Continuity: A function $f(x)$ is continuous at $x = a$ if the following three conditions are met:
  • $f(a)$ is defined.
  • $\lim_{x \to a} f(x)$ exists.
  • $\lim_{x \to a} f(x) = f(a)$.
• 🚧 Discontinuities: Piecewise functions can have discontinuities at the points where the definition changes or where any of the sub-functions are discontinuous. Common types include jump discontinuities (left and right limits exist but are unequal) and removable discontinuities (limit exists but is not equal to the function value).

🧪 Practice Quiz

1. Consider the function: $f(x) = \begin{cases} x^2, & x < 1 \\ 2x, & x \geq 1 \end{cases}$. What is $\lim_{x \to 1^-} f(x)$?

   1. 0
   2. 1
   3. 2
   4. Does not exist

2. Using the same function as above, is $f(x)$ continuous at $x = 1$?

   1. Yes
   2. No
   3. Cannot be determined
   4. Continuous only from the left

3. Let $g(x) = \begin{cases} x + 1, & x < 2 \\ 3, & x = 2 \\ x^2 - 1, & x > 2 \end{cases}$. What is $\lim_{x \to 2} g(x)$?

   1. 3
   2. 4
   3. Does not exist
   4. 5

4. For what value of $k$ is the following function continuous at $x = 3$? $h(x) = \begin{cases} kx, & x \leq 3 \\ x + 6, & x > 3 \end{cases}$

   1. 1
   2. 2
   3. 3
   4. 4

5. Consider the function $p(x) = \begin{cases} \frac{x^2 - 4}{x - 2}, & x \neq 2 \\ 4, & x = 2 \end{cases}$. Is $p(x)$ continuous at $x = 2$?

   1. Yes
   2. No
   3. Cannot be determined
   4. Continuous only from the right

6. Given $q(x) = \begin{cases} x^3, & x < 0 \\ 0, & x = 0 \\ x^2, & x > 0 \end{cases}$. What is $\lim_{x \to 0} q(x)$?

   1. 0
   2. 1
   3. Does not exist
   4. -1

7. The function $r(x) = \begin{cases} \frac{\sin(x)}{x}, & x \neq 0 \\ 1, & x = 0 \end{cases}$ is:

   1. Continuous everywhere
   2. Discontinuous at $x = 0$
   3. Continuous only for $x > 0$
   4. Continuous only for $x < 0$