Solved Examples of Removable Discontinuities in Calculus

📚 Quick Study Guide

• 🔍 A removable discontinuity occurs at a point $x=c$ if $\lim_{x \to c} f(x)$ exists, but $f(c)$ is either undefined or not equal to the limit.
• 💡 To remove a discontinuity, redefine the function $f(x)$ at $x=c$ such that $f(c) = \lim_{x \to c} f(x)$.
• 📝 Common examples include rational functions where both numerator and denominator are zero at $x=c$. These can often be simplified by factoring.
• ➗ Factoring and simplifying expressions is key to finding and removing these discontinuities. Remember your algebra!
• 📈 Graphically, a removable discontinuity appears as a 'hole' in the graph.
• ✍️ To find the limit, try direct substitution first. If you get an indeterminate form (like $\frac{0}{0}$), try factoring, rationalizing, or L'Hôpital's Rule (if applicable).

Practice Quiz

1. What condition must be met for a function $f(x)$ to have a removable discontinuity at $x=a$?
1. A) $f(a)$ exists.
2. B) $\lim_{x \to a} f(x)$ does not exist.
3. C) $\lim_{x \to a} f(x)$ exists but is not equal to $f(a)$.
4. D) $f(x)$ is continuous at $x=a$.
2. Which of the following functions has a removable discontinuity at $x=2$?
1. A) $f(x) = \frac{1}{x-2}$
2. B) $f(x) = \frac{x^2 - 4}{x-2}$
3. C) $f(x) = \sqrt{x-2}$
4. D) $f(x) = \ln(x-2)$
3. What value should $f(2)$ be assigned to remove the discontinuity in $f(x) = \frac{x^2 - 4}{x-2}$?
1. A) 0
2. B) 2
3. C) 4
4. D) Undefined
4. Consider the function $g(x) = \frac{(x-3)(x+1)}{x-3}$. What is the limit of $g(x)$ as $x$ approaches 3?
1. A) 0
2. B) 4
3. C) -1
4. D) The limit does not exist.
5. For the function $h(x) = \frac{x^2 - 5x + 6}{x-2}$, what is the value of $x$ where the removable discontinuity occurs?
1. A) 2
2. B) 3
3. C) -2
4. D) -3
6. If $f(x) = \frac{\sin(x)}{x}$, what value should $f(0)$ be assigned to remove the discontinuity at $x=0$?
1. A) 0
2. B) 1
3. C) Undefined
4. D) $\pi$
7. What is the key step in identifying a removable discontinuity in a rational function?
1. A) Finding the derivative.
2. B) Factoring the numerator and denominator.
3. C) Evaluating the function at $x=0$.
4. D) Finding the y-intercept.