For the function f graphed as follows, approximate each of...

For the function $f$ graphed as follows, approximate each of the limits and values.
$$
\lim _{x \rightarrow 0} f(x), \lim _{x \rightarrow 1} f(x), \lim _{x \rightarrow-\infty} f(x), \text { and } \lim _{x \rightarrow \infty} f(x) .
$$

Key Concepts

Limit of a Function

The limit of a function describes the behavior of the function as the input value approaches a certain point. It is concerned with what value the outputs get arbitrarily close to, regardless of the function’s actual value at that point. Understanding limits is fundamental in calculus as they provide insight into the behavior of functions near points of discontinuity or in situations where the function is not explicitly defined.

One-Sided Limits

One-sided limits examine the behavior of a function as the input approaches a specific value from only one side—either from the left or from the right. This concept is particularly important when a function has different behaviors or does not have the same value when approached from different directions, which can indicate a discontinuity or a jump in the graph.

Limits at Infinity

Limits at infinity investigate the behavior of a function as the input grows without bound—either positively or negatively. This concept is used to determine horizontal asymptotes, helping to describe the end behavior of functions. It tells us the value that the outputs are approaching as the absolute value of the inputs becomes very large.

Continuity

Continuity is the property of a function where, at a given point, the function’s limit as the input approaches that point exists and equals the function’s value at the point. In the context of limits, continuity ensures that there is no disruption or jump in the graph, allowing for a smoother analysis of the function’s behavior near the examined value.

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