Problem 4 - APEX Calculus

Question

T/F: If \(\lim _{x \rightarrow 1} f(x)=5,\) then \(\lim _{x \rightarrow 1^{-}} f(x)=5\)

Step-by-Step Solution

Answer

True, both the left and right limits equal 5.

Step 1: Understanding the Statement

We are given that the limit of function \( f(x) \) as \( x \to 1 \) is 5. We need to determine if the left-hand limit, \( \lim_{x \to 1^-} f(x) = 5 \), follows from this. To do this, let's consider what these limits mean.

Step 2: Limit Definition Review

The limit \( \lim_{x \to 1} f(x) = 5 \) means that as \( x \) approaches 1 from both sides (left and right), the function \( f(x) \) approaches 5. The left-hand limit \( \lim_{x \to 1^-} f(x) = 5 \) only considers \( x \) approaching 1 from values less than 1.

Step 3: Correlation Between Limits

If \( \lim_{x \to 1} f(x) = 5 \), it implies that both \( \lim_{x \to 1^-} f(x) \) and \( \lim_{x \to 1^+} f(x) \) must equal 5 individually, as this is required for the limit at a point to exist.

Step 4: Conclusion

Given that \( \lim_{x \to 1} f(x) = 5 \) means both the left-hand and right-hand limits equal 5, the statement is true: \( \lim_{x \to 1^-} f(x) = 5 \).

Key Concepts

Left-hand Limit

When we discuss the left-hand limit, we focus on the behavior of a function as the input values approach a specific point from the left side, or from values less than the point of interest.
For example, in our exercise, the left-hand limit of function \( f(x) \) as \( x \) approaches 1 is represented by \( \lim_{x \to 1^-} f(x) \).
This notation shows that we are considering only those \( x \) values that are slightly less than 1 and seeing how their corresponding \( f(x) \) values behave as \( x \) gets closer to 1 from that side.

• Definition: The left-hand limit exists if as \( x \to c^- \), \( f(x) \) tends towards a specific value \( L \).
• Importance: It helps in gauging the behavior of a function from a specific direction and is essential for analyzing the full nature of limits.

It's critical to note that knowing one-directional behavior helps in understanding the overall characteristics of the function, especially if combined with right-hand limits.

Right-hand Limit

The right-hand limit is similar to its counterpart but focuses on the values approaching a specific point from the right side, meaning from values greater than the point.
For instance, \( \lim_{x \to 1^+} f(x) \) looks at \( x \) values just greater than 1 and tracks how \( f(x) \) behaves as \( x \) approaches 1 from the right.

• Definition: The right-hand limit of a function \( f(x) \) as \( x \) approaches \( c \) from the right is defined as the value that \( f(x) \) approaches as \( x \to c^+ \).
• Visualization: Imagine creeping up towards the point from the right on a graph and focusing on the y-values.
• Significance: It's a necessary component of evaluating the existence of a bilateral limit, providing the full picture in conjunction with left-hand limits.

By understanding both the left and right, we can say more confidently what happens overall as \( x \to c \).

Continuity

Continuity revolves around whether a function behaves predictably and seamlessly at a particular point.
For a function to be continuous at a point \( c \), the following must hold true:

• The function is defined at \( c \), meaning \( f(c) \) exists.
• The limit as \( x \) approaches \( c \) from both sides exists.
• The left-hand and right-hand limits at \( c \) are equal, and they both equal \( f(c) \).

In our exercise with \( \lim_{x \to 1} f(x) = 5 \), if both left-hand and right-hand limits at 1 equal 5 and \( f(1) = 5 \), then \( f(x) \) is continuous at 1.
Continuity ensures a smooth transition without jumps or holes in the graph at that point.

Limit Definition

The concept of a limit is foundational in calculus, representing the value that a function approaches as the input approaches a certain point.
The general idea is seeing what happens to \( f(x) \) as \( x \) gets infinitely close to a number \( c \).

• Definition: \( \lim_{x \to c} f(x) = L \) if for every number \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - c| < \delta \), then \( |f(x) - L| < \epsilon \).
• Purpose: Helps in understanding behavior around points which actually might not even be defined or clear from the function's perspective.
• Application: Essential for analyzing changes and continuity, solving differential equations, and understanding function behavior globally.

In our scenario, understanding that \( \lim_{x \to 1} f(x) = 5 \) implies all around approach from both left and right culminating on this one value.