Understanding Discontinuities in Functions: Exploring Removable, Jump, and Infinite Discontinuities using Limits

What are the different kinds of discontinuities and how do we describe them using limits?

There are three main types of discontinuities in functions: removable, jump, and infinite discontinuities

There are three main types of discontinuities in functions: removable, jump, and infinite discontinuities. We can describe these discontinuities using the concept of limits.

1. Removable Discontinuity (or Point Discontinuity):
A removable discontinuity occurs when a function has a hole in its graph at a specific point. This means that the function is defined at that point, but the graph has a gap. To describe a removable discontinuity using limits, we check if the limit of the function as it approaches the point equals the value of the function at that point. If yes, then it is a removable discontinuity. Mathematically, we write:

lim(x→a) f(x) = f(a), where a is the x-coordinate of the point.

For example, consider the function f(x) = (x^2 – 4)/(x – 2). At x = 2, there is a removable discontinuity. We can evaluate the limit as x approaches 2:

lim(x→2) f(x) = lim(x→2) (x+2) = 4.

Since f(2) also equals 4, we can say that the discontinuity at x = 2 is removable.

2. Jump Discontinuity:
A jump discontinuity occurs when a function approaches two different values from the left and right sides of a particular point. There is a sudden “jump” in the graph at that point. To describe a jump discontinuity using limits, we check if the left-hand limit and the right-hand limit exist and are different. Mathematically, we write:

lim(x→a^-) f(x) ≠ lim(x→a^+) f(x), where a is the x-coordinate of the point.

For example, consider the function g(x) = |x|. At x = 0, there is a jump discontinuity. We can evaluate the left-hand limit as x approaches 0:

lim(x→0^-) g(x) = -1.

And the right-hand limit as x approaches 0 is:

lim(x→0^+) g(x) = 1.

Since the left-hand limit is -1 and the right-hand limit is 1, we can say that the discontinuity at x = 0 is a jump discontinuity.

3. Infinite Discontinuity:
An infinite discontinuity occurs when the function approaches either positive or negative infinity as it approaches a particular point. To describe an infinite discontinuity using limits, we check if the limit of the function as it approaches the point tends to infinity or negative infinity. Mathematically, we write:

lim(x→a) f(x) = ±∞, where a is the x-coordinate of the point.

For example, consider the function h(x) = 1/x. At x = 0, there is an infinite discontinuity. We can evaluate the limit as x approaches 0:

lim(x→0) h(x) = ±∞.

Since the function tends toward positive or negative infinity as x approaches 0, we can say that the discontinuity at x = 0 is an infinite discontinuity.

These three types of discontinuities, removable, jump, and infinite, help us understand how a function behaves around specific points and help us describe them using limits.

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