Name: oscillating limits near discontinuities
Uploaded: 2025-11-26T05:43:41.000Z
Duration: 1 min 19 s
Channel: JoVE

Consider a bicycle wheel with closely spaced spokes spinning very fast.At the outer edge, the spokes move slowly enough to remain visible individually. Closer to the hub, the spokes flash past more quickly. At the center, the motion is too fast for the human eye to distinguish, and the spokes appear blurred.This spinning wheel provides a physical analogy for the concept of an oscillating discontinuity.It can be studied using the equation, the limit of the function sin&#8289;e one over x as x tends to zero.Just as a point on a spinning wheel never settles into a fixed position, the function fails to converge to a specific value.As the input x approaches zero, one over x grows towards positive or negative infinity.Because the sine function cycles repeatedly between &#8211;1 and +1, the output of sine one over x oscillates between these two values infinitely. The oscillations become increasingly frequent as x nears zero.Since the output never settles on one value, the limit at zero does not exist.

limits with oscillating discontinuities

An oscillating discontinuity is a type of discontinuity in which a function&#8217;s values fluctuate infinitely often as the input approaches a particular point. Unlike jump discontinuities, where the function suddenly shifts between two values, or infinite discontinuities, where the function diverges without bound, an oscillating discontinuity arises from rapid back-and-forth variation. Because the function never stabilizes toward a single value, no finite limit exists at that point.One of the most well-known examples is the function<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:90.08%;'><img style='aspect-ratio:300/20;' src='https://cloudfront.jove.com/files/ftp_upload/41305/41305_Equation_1_editor_41305.svg' alt='41305_Equation_1.svg' /></figure>As x approaches zero, the reciprocal term of the sine function, 1/x, grows without bound. The sine function, being periodic with a fixed range between &#8722;1 and 1, forces the output to oscillate endlessly between these bounds. The closer x gets to zero, the more rapidly the oscillations occur, resulting in an infinite density of fluctuations near the origin. Despite being bounded, the function has no well-defined limit at zero.This phenomenon can be visualized using physical analogies. For instance, a spinning bicycle wheel with closely spaced spokes appears to flash past more rapidly as one looks closer to the hub. Eventually, the motion becomes so fast that the spokes blur together. Similarly, in the mathematical case, the oscillations of the function become so compressed near zero that the function never settles on a single value.Oscillating discontinuities illustrate the subtle difference between boundedness and convergence in analysis. They demonstrate how a function can remain within fixed numerical limits but still fail to possess a limit, highlighting the precision required in calculus when discussing continuity and the behavior of functions near singular points.

Limits with Oscillating Discontinuities

An oscillating discontinuity is a type of discontinuity in which a function&#8217;s values fluctuate infinitely often as the input approaches a particular ...

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Limits

KEY TERMS AND CONCEPTS

oscillating limits near discontinuities

Oscillating discontinuities happen when a function keeps changing value as the input gets close to a point. The output does not settle down to one number. Because of that, the limit at that point does not exist.

This kind of discontinuity is different from a jump discontinuity and an infinite discontinuity. In a jump discontinuity, the function shifts suddenly between two values. In an infinite discontinuity, the function grows without bound.

A well-known example comes from the function involving sine and a reciprocal term near x = 0. As x approaches zero, the reciprocal term grows without bound. At the same time, the sine function stays between -1 and 1, so the output keeps oscillating between those bounds.

The closer x gets to zero, the faster the oscillations happen. The fluctuations become extremely dense near the origin. Even though the function stays bounded, it still has no well-defined limit at zero.

A physical picture can help explain this behavior. A spinning bicycle wheel with closely spaced spokes seems to flash faster as you look closer to the hub. Soon the motion becomes so fast that the spokes blur together.

The math works in a similar way. Near zero, the function changes too quickly to settle on a single value. Oscillating discontinuities show that a function can stay within fixed limits and still fail to converge, which is an important idea in calculus.

Oscillating Limits Near Discontinuities

tangent to a curve

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Secant lines and tangent lines help describe how a curve changes at a chosen point. The curve shown here comes from a function where each output is the square of the input. It bends upward and becomes steeper as you move farther from the center. To study the curve near one position, a second point is chosen slightly ahead on the graph. A straight line through the two points is called a secant line. It crosses the curve and shows the average rise or fall between those points. Because it spans a...

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introduction to limits

Limits in Graphs and Real-World Change

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types of limits i

One-Sided Limits and Limit Failure

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types of limits ii

Vertical Asymptotes and Infinite Limits

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Oscillating discontinuities happen when a function keeps changing value as the input gets close to a point. The output does not settle down to one number. Because of that, the limit at that point does not exist. This kind of discontinuity is different from a jump discontinuity and an infinite discontinuity. In a jump discontinuity, the function shifts suddenly between two values. In an infinite discontinuity, the function grows without bound. A well-known example comes from the function...

limit laws i

Limit Laws for Combining Functions

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limit laws ii

Quotient, Power, and Root Limits

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evaluating limits by direct substitution

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limits at infinity

Horizontal Asymptotes in Function Graphs

Horizontal asymptotes describe how a function behaves as the input grows very large. In this topic, a decreasing function gets smaller and smaller as the input increases. Its output moves closer and closer to one fixed value. The function does not reach that value, but it approaches it without limit. On a graph, this long-term behavior appears as a curve that flattens out. The graph seems to level off near a horizontal line. That line is the horizontal asymptote, which the curve approaches but...

key terms and concepts

Explore oscillating discontinuities, including how rapid sine-based fluctuations near zero prevent a limit from existing, with a bicycle wheel analogy.

Oscillating Limits Near Discontinuities | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Oscillating Limits Near Discontinuities

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