11.15: Limits at Infinity

The function that decreases as the input becomes very large provides a clear example of how mathematical functions can behave at extreme values. When the input increases continuously, the output becomes smaller and smaller, getting closer to a particular fixed value. Although the output never actually reaches this value, it moves nearer to it without limit. This behavior is a fundamental concept in understanding how functions behave as the input grows indefinitely. The graphical representation of this function shows that its curve becomes flatter and appears to level off near a specific horizontal line. This line, which the graph approaches but never touches, represents the long-term behavior of the function.

Another example of this type of behavior is seen in a function that gradually levels off at different values depending on whether the input increases or decreases without limit. The output approaches a certain upper value as the input becomes extremely large in the positive direction. Conversely, as the input becomes very large in the negative direction, the output moves closer to a different lower value. These two boundaries are not crossed or reached by the function but indicate how the function behaves as the input moves toward extreme values in either direction. On a graph, the function’s curve bends gently and gets closer to these upper and lower boundaries, showing a smooth transition without sudden changes.

This type of analysis helps understand how functions behave in the long run and is essential in many areas of mathematics, especially in describing trends, estimating values, and modeling real-world phenomena in abstract terms. These features are significant when analyzing the stability or limits of systems represented by mathematical functions.

Limits of a function can be evaluated as x approaches positive or negative infinity. These two limits are distinct and must be checked separately.

Consider the function x cubed. As x approaches positive infinity, the value increases without bound.

As x approaches negative infinity, the value decreases without bound.

In contrast, the sine function oscillates between −1 and 1. Since it never settles, its limit at infinity does not exist.

Some functions approach a finite value, such as one divided by x plus 2. As x tends to infinity, one over x becomes zero, leaving the value 2. This horizontal line, y equals 2, is called a horizontal asymptote.

This concept appears in real circuits, such as when a capacitor is charged in a series RC circuit.

When a battery is connected, the charge on the capacitor increases with time. Taking the limit as time t approaches infinity, the exponential term will be zero, and the capacitor’s charge will approach a constant maximum value, which represents the horizontal asymptote of the curve.