Problem 38 - Brief Applied Calculus

Question

For each piecewise linear function, find: a. \(\lim _{x \rightarrow 4^{-}} f(x)\) b. \(\lim _{x \rightarrow 4^{+}} f(x)\) c. \(\lim _{x \rightarrow 4} f(x)\) $$ f(x)=\left\\{\begin{array}{ll} 5-x & \text { if } x<4 \\ 2 x-5 & \text { if } x \geq 4 \end{array}\right. $$

Step-by-Step Solution

Answer

a. 1 b. 3 c. The limit does not exist.

Step 1: Identify the function pieces

The function is defined as a piecewise function with two parts: 1. If \( x < 4 \), \( f(x) = 5 - x \).2. If \( x \geq 4 \), \( f(x) = 2x - 5 \).

Step 2: Finding \( \lim_{x \to 4^-} f(x) \)

To find the left-hand limit as \( x \to 4^- \), we use the part of the function defined for \( x < 4 \), which is \( 5 - x \). Substitute \( x \) with values approaching 4 from the left:\( \lim_{x \to 4^-} (5 - x) = 5 - 4 = 1 \).

Step 3: Finding \( \lim_{x \to 4^+} f(x) \)

To find the right-hand limit as \( x \to 4^+ \), we use the part of the function defined for \( x \geq 4 \), which is \( 2x - 5 \). Substitute \( x \) with values approaching 4 from the right:\( \lim_{x \to 4^+} (2x - 5) = 2(4) - 5 = 8 - 5 = 3 \).

Step 4: Finding \( \lim_{x \to 4} f(x) \)

To find \( \lim_{x \to 4} f(x) \), both the left-hand and right-hand limits must be equal at \( x = 4 \).Previously, we found:- \( \lim_{x \to 4^-} f(x) = 1 \)- \( \lim_{x \to 4^+} f(x) = 3 \)Since they are not equal, \( \lim_{x \to 4} f(x) \) does not exist.

Key Concepts

Limits

Limits are a fundamental concept in calculus, especially when dealing with piecewise functions. In simple terms, a limit describes the value that a function approaches as the input approaches a certain point. In the context of the given exercise:

• For a piecewise function, like the one given, limits help us understand how the function behaves near the point where the definition of the function changes.
• The notation \( \lim_{x \to 4^-} \) means approaching 4 from the left (values less than 4), while \( \lim_{x \to 4^+} \) means approaching from the right (values greater than 4).

By evaluating both left-hand and right-hand limits, you can check for continuity at a particular point, revealing whether the function presents any jump or break at \( x = 4 \). In our example, the limits did not match and were therefore crucial in determining that the overall limit \( \lim_{x \to 4} f(x) \) does not exist.

Continuous Functions

A function is continuous at a point if there is no interruption in its graph at that point. For piecewise functions, establishing continuity involves checking if the function’s pieces connect smoothly where they meet. Here's what this entails:

• The function must be defined at that point, which means \( f(x) \) must have a value at \( x \).
• The left-hand limit and right-hand limit as \( x \) approaches the point must be equal.
• The value of the function at that point must also be equal to these limits.

A function is said to be discontinuous if any of these conditions are not met. In the case of our function \( f(x) \), since \( \lim_{x \to 4^-} f(x) eq \lim_{x \to 4^+} f(x) \), the function is not continuous at \( x = 4 \). Understanding continuity in piecewise functions helps in analyzing and graphing them correctly, ensuring that all transitions between their parts are smooth.

Calculus

Calculus is the branch of mathematics that studies how things change and is foundational for understanding limits and continuity. It provides tools—like differentiation and integration—that are essential in analyzing functions. Here's how calculus relates to our discussion:

• It allows us to calculate limits precisely as \( x \) approaches any point, either from the left or the right, using algebraic techniques.
• Continuity is assessed using calculus concepts, ensuring that there are no abrupt changes in the behavior of the function at specific points.
• Piecewise functions in calculus often model real-world scenarios where different rules apply at different stages or intervals.

This problem, showing how calculus addresses changes in function definitions through limits and continuity, is just the start of much deeper explorations into how functions behave. For students, grasping these basics sets the stage for tackling more complex calculus challenges.