Question

Use the following definitions. Assume fexists for all \(x\) near a with \(x>\) a. We say the limit of \(f(x)\) as \(x\) approaches a from the right of a is \(L\) and write \(\lim _{x \rightarrow a^{+}} f(x)=L,\) if for any \(\varepsilon>0\) there exists \(\delta>0\) such that $$|f(x)-L|<\varepsilon \quad \text { whenever } \quad 0< x-a<\delta$$ Assume fexists for all \(x\) near a with \(x < \) a. We say the limit of \(f(x)\) as \(x\) approaches a from the left of a is \(L\) and write \(\lim _{x \rightarrow a^{-}} f(x)=L,\) if for any \(\varepsilon > 0 \) there exists \(\delta > 0\) such that $$|f(x)-L| < \varepsilon \quad \text { whenever } \quad 0< a-x <\delta$$ Why is the last inequality in the definition of \(\lim _{x \rightarrow a} f(x)=L,\) namely, \(0<|x-a|<\delta,\) replaced with \(0

Short answer

Answer: The inequality changes for the right-hand limit definition to ensure that we only consider values of 'x' greater than 'a', as we are concerned with the behavior of the function as it approaches 'a' from the right side. In the general limit definition, the absolute value in the inequality allows for both left and right sides surrounding the point 'a', while in the left-hand limit definition, we only consider values of 'x' less than 'a' by changing the inequality accordingly.

Step-by-step solution

Understand the General Definition of Limit

The general definition of the limit states that for \(\lim_{x \rightarrow a} f(x) = L\), \(|f(x) - L| < \varepsilon\) whenever \(0 < |x - a| < \delta\). Here, we consider the behavior of the function as it approaches 'a' from both directions (i.e., left and right).

Recognize the Right-Hand Limit Definition

Now, let's focus on the right-hand limit definition: \(\lim_{x \rightarrow a^{+}} f(x) = L\). In this case, we are only concerned with the behavior of the function as it approaches 'a' from the right side (i.e., \(x > a\)). This is why the inequality changes from \(0 < |x - a| < \delta\) to \(0 < x - a < \delta\). This ensures that we only consider values of 'x' greater than 'a'.

Recognize the Left-Hand Limit Definition

Similarly, in the left-hand limit definition, we are only concerned with the behavior of the function as it approaches 'a' from the left side (i.e., \(x < a\)). Therefore, the inequality changes to \(0 < a - x < \delta\) to ensure that we only consider values of 'x' less than 'a'.

Explain the Difference Between the Inequalities

In the general limit definition, the absolute value in the inequality \(0 < |x - a| < \delta\) allows us to consider both left and right sides surrounding the point 'a'. However, in the right-hand and left-hand limit definitions, we only want to consider the behavior of the function from one side (either right or left). Therefore, we change the inequality to \(0 < x - a < \delta\) for the right-hand limit and \(0 < a - x < \delta\) for the left-hand limit, respectively, to restrict our consideration to one specific side of the point 'a'.

Key concepts

Right-Hand Limit

The right-hand limit is a concept that helps us understand how a function behaves as it gets closer to a specific point from the right side on the graph. This is especially useful when analyzing functions that might behave differently from different directions. When we write \( \lim_{x \rightarrow a^{+}} f(x) = L \), it's a concise way of saying, "As \( x \) approaches \( a \) from values greater than \( a \), the function \( f(x) \) gets closer and closer to \( L \)."

Here's what makes the right-hand limit definition special:

• We focus exclusively on points where \( x > a \), meaning we're looking at the function approaching from the right side of \( a \).
• The inequality \( 0 < x - a < \delta \) is used to describe this situation, ensuring that only values to the right of \( a \) are included.
• This approach helps isolate the behavior of the function in one specific direction, which can be crucial in piecewise functions or where there’s a possibility of different limits from different sides.

Understanding right-hand limits is fundamental, as they are a stepping stone to more complex concepts, such as continuity and differentiability.

Left-Hand Limit

The left-hand limit is another side of the limit definition, focusing on what happens as \( x \) approaches a point from the left side of the graph. This mirrors the right-hand limit but from the opposite direction. When you see \( \lim_{x \rightarrow a^{-}} f(x) = L \), it indicates that "As \( x \) approaches \( a \) from values less than \( a \), the function \( f(x) \) intends to get closer to \( L \)."

Key points about left-hand limits include:

• The focus is solely on \( x < a \), capturing the function's behavior as it nears from the left.
• It uses the inequality \( 0 < a - x < \delta \), precisely selecting the region to the left of \( a \).
• This is important in cases where the function could have different behaviors or discontinuities from the left compared to the right.

Grasping left-hand limits is crucial, especially in deeper mathematical analysis, and helps complete the picture when examining function behaviors around specific points.

Inequality in Limits

Inequalities play a fundamental role in understanding how limits that approach from one direction, whether right or left, differ from general limits. Generally, the limit definition involves \( 0 < |x - a| < \delta \), which considers both sides of \( a \): the right and the left together.

For directional limits:

• The right-hand limit and its inequality \( 0 < x - a < \delta \) ensures that we're examining points towards the right.
• The left-hand limit uses \( 0 < a - x < \delta \) to focus on points approaching from the left.
• These specific inequalities provide the precision needed to assess functions that may not reach the same limit from both sides, aiding in identifying potential asymmetries or discontinuities.

By refining our focus with these inequalities, we gain a deeper understanding of the function's behavior. This is especially beneficial in calculus, where the precise behavior of functions around points determines continuity, derivatives, and integrals.