๐ Understanding Removable Discontinuities
A removable discontinuity, sometimes called a hole, occurs in a rational function when a factor cancels out from both the numerator and the denominator. This cancellation creates a point where the function is undefined, but the limit exists at that point. Let's dive in!
๐ History and Background
The concept of discontinuities became crucial with the formalization of calculus in the 17th and 18th centuries. Mathematicians like Leibniz and Newton grappled with defining functions and their behavior, leading to the identification and classification of different types of discontinuities. Removable discontinuities are among the simplest to understand, as they represent a single point of exclusion that can often be 'patched' by redefining the function at that point.
๐ Key Principles
- ๐ Factorization: Start by factoring both the numerator and the denominator of the rational function.
- ๐ซ Cancellation: Identify any common factors that can be canceled from both the numerator and the denominator.
- ๐ Identifying the Hole: Set the canceled factor equal to zero and solve for $x$. This $x$ value represents the location of the removable discontinuity.
- ๐ Finding the $y$-value: Substitute the $x$ value (from the previous step) into the simplified function (after cancellation) to find the corresponding $y$ value. This gives you the coordinates of the hole $(x, y)$.
- โ๏ธ Limit Existence: The limit of the function as $x$ approaches the $x$-value of the hole exists, even though the function is not defined at that specific point.
โ Example 1: A Simple Rational Function
Consider the function: $f(x) = \frac{x^2 - 4}{x - 2}$
- ๐ก Factorization: Factor the numerator: $x^2 - 4 = (x - 2)(x + 2)$
- ๐ซ Cancellation: Rewrite the function: $f(x) = \frac{(x - 2)(x + 2)}{x - 2}$. Cancel the common factor $(x - 2)$.
- ๐ Hole Location: Set the canceled factor equal to zero: $x - 2 = 0$, so $x = 2$.
- ๐งฉ $y$-value: Substitute $x = 2$ into the simplified function $x + 2$: $2 + 2 = 4$.
Therefore, there is a removable discontinuity (a hole) at the point $(2, 4)$.
๐ Example 2: A More Complex Function
Consider the function: $g(x) = \frac{x^2 + 5x + 6}{x^2 + x - 2}$
- ๐ก Factorization: Factor both numerator and denominator:
$x^2 + 5x + 6 = (x + 2)(x + 3)$ and $x^2 + x - 2 = (x + 2)(x - 1)$
- ๐ซ Cancellation: Rewrite the function: $g(x) = \frac{(x + 2)(x + 3)}{(x + 2)(x - 1)}$. Cancel the common factor $(x + 2)$.
- ๐ Hole Location: Set the canceled factor equal to zero: $x + 2 = 0$, so $x = -2$.
- ๐งฉ $y$-value: Substitute $x = -2$ into the simplified function $\frac{x + 3}{x - 1}$: $\frac{-2 + 3}{-2 - 1} = \frac{1}{-3} = -\frac{1}{3}$.
Therefore, there is a removable discontinuity at the point $(-2, -\frac{1}{3})$.
๐ Real-world Examples
- โ๏ธ Engineering Design: In designing a bridge, engineers might model the load-bearing capacity as a rational function. A removable discontinuity could represent a theoretical flaw in the design that is 'patched' by reinforcing that specific point.
- ๐งช Chemical Reactions: When modeling reaction rates, a removable discontinuity might represent a condition where a specific intermediate step is bypassed, leading to a slightly different overall reaction.
- ๐ Economic Models: In supply and demand models, a removable discontinuity could represent a price point where a temporary market inefficiency exists, but the market quickly corrects itself.
๐ Conclusion
Removable discontinuities are essentially 'holes' in the graph of a rational function. Identifying them involves factoring, canceling common factors, and then finding the coordinates of the hole. Understanding removable discontinuities is crucial for analyzing the behavior of rational functions and their applications in various fields.