Problem 88 - Calculus: Early Transcendentals

Question

Finding a constant Suppose $$f(x)=\left\\{\begin{array}{ll} 3 x+b & \text { if } x \leq 2 \\ x-2 & \text { if } x>2 \end{array}\right.$$ Determine a value of the constant \(b\) for which \(\lim _{\vec{F} \rightarrow^{2}} f(x)\) exists and state the value of the limit, if possible.

Step-by-Step Solution

Answer

Answer: The value of the constant b for which the limit of the function f(x) as x approaches 2 exists is b = -6, and the value of the limit is 0.

Step 1: Identify the function's expressions defined for each interval

The given function f(x) is a piecewise function, with two expressions: 1. f(x) = 3x + b, when x ≤ 2 2. f(x) = x - 2, when x > 2

Step 2: Find the limit of f(x) as x approaches 2 from the left

We evaluate the limit of the first expression as x approaches 2 from the left (x < 2): $$\lim_{x \to 2^{-}} f(x) = \lim_{x \to 2^{-}} (3x + b)$$ Since this expression does not have any denominator approaching zero, we can directly substitute x = 2: $$\lim_{x \to 2^{-}} f(x) = 3(2) + b = 6 + b$$

Step 3: Find the limit of f(x) as x approaches 2 from the right

Next, we evaluate the limit of the second expression as x approaches 2 from the right (x > 2): $$\lim_{x \to 2^{+}} f(x) = \lim_{x \to 2^{+}} (x - 2)$$ Again, we can simply substitute x = 2 as there are no denominators approaching zero: $$\lim_{x \to 2^{+}} f(x) = 2 - 2 = 0$$

Step 4: Equate the left and right limits to find the value of b

To find the value of b for which the limit exists, we need to equate the left and right limits: $$6 + b = 0$$ Solving this equation, we obtain: $$b = -6$$

Step 5: Determine the limit of the function

Now that we have the value of b, we can also find the limit of the function when it exists. Since the left and the right limits are equal when b = -6, the limit of the function as x approaches 2 is: $$\lim_{x \to 2} f(x) = 6 + (-6) = 0$$ Thus, we have found the value of b for which the limit exists, b = -6, and the value of the limit is 0.

Key Concepts

Understanding Piecewise Functions

Piecewise functions are a type of function that have different expressions or rules for different parts of their domain. They are defined piece by piece, meaning for different intervals of the input variable, a different rule applies. This concept can be really useful when a single formula cannot describe a function's entire behavior.

For example, consider the function given in the exercise:

• For values of \(x\) less than or equal to 2, the function is defined by \(f(x) = 3x + b\).
• For values of \(x\) greater than 2, the function is defined by \(f(x) = x - 2\).

Each piece responds to different intervals of \(x\). The challenge often with piecewise functions, like in this exercise, is ensuring that they are consistent at the points where the pieces meet. This involves checking limits and ensuring continuity, especially at the boundary points.

Achieving Continuity in Piecewise Functions

Continuity of a piecewise function is significant for understanding how smoothly the function behaves at the points where different pieces meet. A function is continuous at a point if there is no interruption; its graph can be drawn without lifting the pen off the paper.

For a piecewise function to be continuous at a point where two pieces meet, the left-hand limit and the right-hand limit must be equal at that point. This is shown in the exercise where we find:

• The left-hand limit as \(x \to 2^-\) is \(6 + b\).
• The right-hand limit as \(x \to 2^+\) is \(0\).

To ensure continuity at \(x = 2\), these limits should equal, which leads to the equation \(6 + b = 0\). Solving this gives \(b = -6\), as the value needed for the piecewise function to be continuous at that boundary point.

Calculus and Limits in Evaluating Functions

Calculus, especially the concept of limits, plays a crucial role in evaluating piecewise functions to determine things like continuity. Limits help us understand how a function behaves as it approaches a certain point, even if it doesn't reach that point.

In our exercise, limits are essential to determine if \(f(x)\) is continuous at \(x = 2\). Calculating limits involves finding how the function behaves as \(x\) gets infinitely close to a specific value from both sides:

• Left-hand limit: \(\lim_{x \to 2^-} (3x + b)\)
• Right-hand limit: \(\lim_{x \to 2^+} (x - 2)\)

These limits are equated to find \(b\) for continuity, showcasing calculus' power to handle complex functions. By understanding these fundamental concepts of calculus, students can solve problems involving limits and continuity more effectively.