91Ó°ÊÓ

One-sided limits in calculus refer to the behavior of a function as its input approaches a particular value from one side only, either the left or the right. This concept allows us to observe the directional tendencies of a function near a specific point. For instance, when exploring the left-sided limit, denoted by \( \lim_{x \to c^-} f(x) \), we're interested in what happens as \( x \) approaches the value \( c \) from values less than \( c \) (the left). Similarly, the right-sided limit, noted as \( \lim_{x \to c^+} f(x) \), examines how the function behaves as \( x \) nears \( c \) from values greater than \( c \) (the right). Understanding the distinct behaviors of these one-sided limits can help in analyzing whether the function is approaching a finite number, positive or negative infinity, or if it doesn't settle to any particular limit. For example, in the exercise, the left-sided limit at \( x = -2 \) evaluated to \(-9\), and so did the right-sided limit.

A two-sided limit is what most people commonly refer to when they talk about limits. It queries how a function behaves as its input gets infinitely close to a specific value from both directions, left and right. It's written as \( \lim_{x \to c} f(x) \), and it exists only when both the left-sided and right-sided limits at \( c \) are equal. This convergence tells us that no matter from which direction we approach, the function consistently targets a single value. This is crucial for determining continuity at a point. If both one-sided limits equal a real number \( L \). The two-sided limit exists and is equal to \( L \). In our exercise, we determined the one-sided limits to be \(-9\) from both sides of \( x = -2 \), establishing that the two-sided limit at this point indeed exists and equals \(-9\).

Evaluating limits involves identifying how a function behaves as it nears a specific point or becomes infinitely large. Here are some helpful pointers:

• **Understand One-Sided Limits:** Always consider whether you're asked for a one-sided limit (left or right) or a full two-sided limit. This helps narrow down which part of the function you're dealing with.
• **Substitution Might Work:** For simple functions, try to substitute the value of interest directly into the function. If the function doesn’t become undefined, the substitute value is the limit.
• **Piecewise Functions:** For functions like in the exercise, evaluate the defining expressions separately for each domain (before, on, and after the point of interest). This way, you can ascertain the one-sided and two-sided limits effectively.

In our example, recognizing that the left and right approaches yielded the same result, ensured that we correctly evaluated that the two-sided limit exists and has a clear value. Safely evaluating limits aids in determining continuity and aids in further investigations of functions.