91Ó°ÊÓ

When dealing with limits, the right-hand limit is an important concept. This refers to the limit of a function as the variable approaches a certain value from the right side, which means considering values greater than the point of interest. In mathematical notation, the right-hand limit of a function \( f(x) \) as \( x \) approaches \( a \) from the right is written as \( \lim_{x \to a^+} f(x) = L \).

To determine this limit, we assume \( x \) can be slightly greater than \( a \) and check if \( f(x) \) tends to a number \( L \) for values of \( x \) close to \( a \). We achieve this by using the \( \varepsilon - \delta \) definition:

• Choose any small positive number \( \varepsilon \).
• Find a \( \delta > 0 \) such that \( |f(x) - L| < \varepsilon \) whenever \( 0 < x - a < \delta \).

In the exercise, when proving the limit \( \lim_{x \to 0^+} f(x) = -4 \), we use the function part \( 2x - 4 \) valid for \( x \geq 0 \). This means that any small positive deviation from zero confirms the function's limit approaching \(-4\) from the right.

The left-hand limit focuses on approaching a particular point from values less than the point itself. When finding the left-hand limit of a function \( f(x) \) as \( x \) approaches \( a \) from the left, it is denoted as \( \lim_{x \to a^-} f(x) = L \).

This concept is pivotal when examining the behavior of a function as \( x \) gets closer to \( a \) from the left side. Like the right-hand limit, it uses the \( \varepsilon - \delta \) approach:

• Select an arbitrary small positive \( \varepsilon \).
• Determine a \( \delta > 0 \) ensuring that \( |f(x) - L| < \varepsilon \) whenever \( 0 < a - x < \delta \).

In the given exercise, to show \( \lim_{x \to 0^-} f(x) = -4 \), we use \( 3x - 4 \) defined for \( x < 0 \). This helps observe the function behaving consistently with \(-4\) as \( x \) nears zero from the left, ensuring that \( f(x) \) approaches \(-4\) accurately through this approach.

The two-sided limit evaluates how a function behaves as it approaches a point from both the left and right sides simultaneously. For a two-sided limit to exist at a point \( a \), both the right-hand limit \( \lim_{x \to a^+} f(x) \) and the left-hand limit \( \lim_{x \to a^-} f(x) \) must exist and be equal. This is usually denoted simply as \( \lim_{x \to a} f(x) = L \).

Understanding this concept is crucial, allowing one to understand whether a function is continuous at a specific point or if there is a common behavior from both directions.

• Confirm the existence and equality of both one-sided limits.
• The two-sided limit \( \lim_{x \to a} f(x) = L \) exists if these conditions are met.

In the scenario of the exercise, confirmation of \( \lim_{x \to 0} f(x) = -4 \) is achieved because both one-sided limits \( \lim_{x \to 0^+} f(x) = -4 \) and \( \lim_{x \to 0^-} f(x) = -4 \) are equal, signifying the function's value approaches \(-4\) regardless of the direction from which \( x \) approaches zero.