Sketch the graphs of functions that have the given limits...

Sketch the graphs of functions that have the given limits and values. (There are multiple correct answers.)
$$
\lim _{x \rightarrow 0^{+}} f(x)=\infty \text { and } \lim _{x \rightarrow 0^{-}} f(x)=\infty
$$

Key Concepts

One-Sided Limits

One-sided limits describe the behavior of a function as the variable approaches a specific value from one direction (either from the left or from the right). They are essential for understanding how a function behaves near a point where its overall limit might not exist or might differ from each side. In this context, the limits as x approaches 0 from both the positive and negative sides are considered separately.

Infinite Limits

An infinite limit occurs when the values of a function increase or decrease without bound as the input approaches a specific point. This notion implies that the function does not approach a finite value, and instead its values become arbitrarily large in magnitude. Here, the function is designed to have values that tend to infinity as x approaches 0 from either side.

Vertical Asymptotes

A vertical asymptote is a line, typically x = a, where a function’s values increase or decrease without bound as x approaches a. When both one-sided limits at x = 0 are infinite, this indicates that there is a vertical asymptote at x = 0, which is a common behavior in functions with unbounded limits at a point.

Graphing Functions with Asymptotic Behavior

Graphing functions that exhibit asymptotic behavior involves depicting how the function behaves as it approaches points where it is not defined, such as vertical asymptotes. Understanding the behavior of one-sided and infinite limits is crucial for correctly sketching these graphs, ensuring that the function reflects the tendency to grow without bound near the asymptote.

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