What is an Infinite Discontinuity? Definition & Examples for Calculus Students

📚 Quick Study Guide

🔍 Infinite Discontinuity: Occurs at a point $c$ where the limit of the function $f(x)$ approaches infinity ($\infty$) or negative infinity $(-\infty)$ as $x$ approaches $c$ from either the left or the right.

💡 Vertical Asymptote: Infinite discontinuities are often associated with vertical asymptotes on the graph of the function. The line $x = c$ is a vertical asymptote if $\lim_{x \to c^+} f(x) = \pm \infty$ or $\lim_{x \to c^-} f(x) = \pm \infty$.

📝 Examples: Functions like $f(x) = \frac{1}{x-2}$ and $f(x) = \tan(x)$ have infinite discontinuities. For $f(x) = \frac{1}{x-2}$, there's an infinite discontinuity at $x = 2$. For $f(x) = \tan(x)$, there are infinite discontinuities at $x = \frac{\pi}{2} + n\pi$, where $n$ is an integer.

📈 Graphical Representation: The graph of a function with an infinite discontinuity will approach a vertical line (the vertical asymptote) without ever touching it.

Practice Quiz

1. Question 1: Which of the following functions has an infinite discontinuity at $x=0$?
   1. $f(x) = x^2$
   2. $f(x) = \frac{1}{x}$
   3. $f(x) = \sin(x)$
   4. $f(x) = x + 1$
2. Question 2: What is the limit of $f(x) = \frac{1}{(x-3)^2}$ as $x$ approaches 3?
   1. 0
   2. 1
   3. $\infty$
   4. Does not exist
3. Question 3: At what value of $x$ does the function $f(x) = \frac{1}{x+5}$ have an infinite discontinuity?
   1. $x = 5$
   2. $x = -5$
   3. $x = 0$
   4. $x = 1$
4. Question 4: Which of the following functions does NOT have an infinite discontinuity?
   1. $f(x) = \frac{1}{x^2}$
   2. $f(x) = \ln(x)$
   3. $f(x) = x^3 + 2x - 1$
   4. $f(x) = \csc(x)$
5. Question 5: The graph of a function with an infinite discontinuity typically has a:
   1. Horizontal asymptote
   2. Vertical asymptote
   3. Removable discontinuity
   4. Hole
6. Question 6: Consider the function $f(x) = \frac{1}{\sin(x)}$. Where does it have infinite discontinuities?
   1. $x = n\pi$, where $n$ is an integer
   2. $x = (2n+1)\frac{\pi}{2}$, where $n$ is an integer
   3. $x = 0$ only
   4. Nowhere
7. Question 7: What happens to the value of $f(x) = \frac{1}{x-a}$ as $x$ approaches $a$ from the left ($x \to a^-$)?
   1. $f(x)$ approaches $\infty$
   2. $f(x)$ approaches $-\infty$
   3. $f(x)$ approaches 0
   4. The limit does not exist