Solved examples of all types of discontinuities in Pre-Calculus.

📚 Quick Study Guide

• 🔍 A discontinuity occurs at a point where a function is not continuous.
• 📈 A removable discontinuity (also known as a hole) can be 'removed' by redefining the function at that point. It often appears when there's a common factor in the numerator and denominator of a rational function.
• 🚧 An infinite discontinuity (also known as a vertical asymptote) occurs when the function approaches infinity (or negative infinity) as $x$ approaches a certain value. For rational functions, this often happens where the denominator is zero.
• 🤸 A jump discontinuity occurs when the function 'jumps' from one value to another at a certain point. The left-hand limit and right-hand limit exist but are not equal.
• 💡To identify discontinuities, look for: division by zero, piecewise functions with differing values at the transition points, and functions like $\tan(x)$ or $\ln(x)$ that have inherent discontinuities.
• 📝 Formal Definition of Continuity: A function $f(x)$ is continuous at $x = a$ if and only if: (1) $f(a)$ is defined, (2) $\lim_{x \to a} f(x)$ exists, and (3) $\lim_{x \to a} f(x) = f(a)$.

🧪 Practice Quiz

1. Question 1: What type of discontinuity is present in the function $f(x) = \frac{x^2 - 4}{x - 2}$ at $x = 2$?
1. A) Jump Discontinuity
2. B) Infinite Discontinuity
3. C) Removable Discontinuity
4. D) No Discontinuity
2. Question 2: Which of the following functions has an infinite discontinuity at $x = 3$?
1. A) $f(x) = \frac{x - 3}{x + 3}$
2. B) $f(x) = \frac{x + 3}{x - 3}$
3. C) $f(x) = x^2 - 9$
4. D) $f(x) = \frac{1}{x + 3}$
3. Question 3: Consider the piecewise function: $f(x) = \begin{cases} x + 1, & x < 0 \\ x^2, & x \geq 0 \end{cases}$. What type of discontinuity, if any, exists at $x = 0$?
1. A) Removable Discontinuity
2. B) Jump Discontinuity
3. C) Infinite Discontinuity
4. D) Continuous
4. Question 4: The function $f(x) = \tan(x)$ has discontinuities at:
1. A) $x = n\pi$, where $n$ is an integer
2. B) $x = \frac{(2n + 1)\pi}{2}$, where $n$ is an integer
3. C) $x = 0$
4. D) $x = \infty$
5. Question 5: What condition must be met for a function $f(x)$ to be continuous at $x = a$?
1. A) $\lim_{x \to a} f(x)$ must exist.
2. B) $f(a)$ must be defined.
3. C) $\lim_{x \to a} f(x) = f(a)$
4. D) All of the above
6. Question 6: Which of the following functions has a removable discontinuity at x = -1?
1. A) $f(x) = \frac{x+1}{x-1}$
2. B) $f(x) = \frac{x-1}{x+1}$
3. C) $f(x) = \frac{x^2 - 1}{x + 1}$
4. D) $f(x) = \frac{1}{x+1}$
7. Question 7: The function $f(x) = \ln(x)$ has a discontinuity at:
1. A) $x = 1$
2. B) $x = 0$
3. C) $x = -1$
4. D) $x = \infty$