Limits at Infinity

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.