91Ó°ÊÓ

In calculus, a limit helps us understand the behavior of a function as the input gets closer to a particular point. When we say the limit of a function \( f(x) \) as \( x \) approaches \( a \), mathematically expressed as \( \lim_{x \to a} f(x) \), it means we are interested in what number \( f(x) \) is getting closer to as \( x \) gets closer and closer to \( a \). It's a fundamental concept that describes how functions behave near specific points.
To determine the existence of a limit, we need to see if the function approaches some finite number from both sides of \( a \). If it does not, then the limit doesn’t exist. It serves as a check if output values of \( f(x) \) become arbitrarily close to a single value, which indicates the point \( a \) is approachable from the function.

Some essential features of limits include:

• The limit focuses only on the behavior close to \( a \), not necessarily what happens at \( a \) itself.
• Limits help in defining derivatives and integrals, crucial tools in calculus.
• Limits maintain consistency whether they involve infinity or finite values, giving insights into infinite behaviors.

When dealing with limits, there are situations where we specifically examine the behavior of a function as it approaches a point from one side only. This is where one-sided limits come into play.

The left-side limit, denoted \( \lim_{x \to a^-} f(x) \), focuses on what \( f(x) \) approaches as \( x \) comes closer to \( a \) from values smaller than \( a \). Conversely, the right-side limit, \( \lim_{x \to a^+} f(x) \), examines the behavior of \( f(x) \) as \( x \) draws nearer to \( a \) from numbers greater than \( a \).

These limits are particularly valuable in situations such as:

• Checking for continuity, where the function may behave differently on each side of the point.
• Determining whether a function jumps or is discontinuous at a particular location.
• Analyzing piecewise functions, which might have different expressions on either side of certain points.

By studying one-sided limits, we gain a more nuanced understanding of the function’s behavior in potentially complex situations.

A two-sided limit considers both one-sided limits simultaneously. For the two-sided limit \( \lim_{x \to a} f(x) \) to exist at a point \( a \), both the left-side limit and the right-side limit must not only exist but also be equal to each other. This ensures that as \( x \) approaches \( a \) from either direction, the function \( f(x) \) consistently approaches the same value.

This requirement plays a crucial role in:

• Determining the overall limit existence: If \( \lim_{x \to a^-} f(x) \) and \( \lim_{x \to a^+} f(x) \) differ, \( \lim_{x \to a} f(x) \) cannot exist.
• Ensuring function continuity: A function can only be continuous at a point if its two-sided limit equals its function value there.
• Simplifying understanding of function behavior across \( x \) values: Two-sided limits provide a comprehensive view of how a function behaves around \( a \), integrating viewpoints from both sides.

Therefore, for the statement in the exercise, once the two-sided limit \( \lim_{x \to a} f(x) \) is confirmed to exist, it implies that both \( \lim_{x \to a^-} f(x) \) and \( \lim_{x \to a^+} f(x) \) must exist, thereby supporting the assertion that the statement is true. By understanding both sides, we comprehend the overall trends in a function at certain critical points.