Types of Limits II

An infinite limit appears on a graph when a function’s output increases or decreases without bound as the input approaches a specific value.

Mathematically, this means that the limit of f(x) increases toward infinity or decreases toward negative infinity as x approaches a.

This behavior indicates the presence of a vertical asymptote—a vertical line that the function approaches but never touches or crosses.

For example, when x equals 2, the function is undefined. But as x approaches 2 from either side, the function’s values rise rapidly without bound. On the graph, this appears as a steep climb, forming a vertical asymptote at x = 2.

A helpful analogy is a rock climber approaching a cliff. The path begins with a gentle slope, but as the rock climber moves further, the incline steepens rapidly.

Near the edge, the slope turns nearly vertical—too steep to continue.

The graph of this scenario reveals a curve that climbs ever more steeply, mirroring how, near a vertical asymptote, a function’s values surge beyond control.