Problem 68 - Calculus Early Transcendentals

Question

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 \(\lim _{x \rightarrow 2} f(x)\) exists and state the value of the limit, if possible.

Step-by-Step Solution

Answer

Function: $$ f(x) = \begin{cases} 3x + b, & x \leq 2 \\ x - 2, & x > 2 \end{cases} $$ Solution: The value of constant b for which the limit of f(x) exists as x approaches 2 is -6.

Step 1: Find the left limit of f(x) as x approaches 2

To find the left limit, we need to consider the function when \(x \leq 2\). $$ \lim_{x \rightarrow 2^{-}} f(x) = \lim_{x \rightarrow 2^{-}} (3x + b) $$ Since the function is continuous for \(x \leq 2\), we can evaluate the limit directly by substitution: $$ \lim_{x \rightarrow 2^{-}} f(x) = 3(2)+b = 6+b $$

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

To determine the right limit, we need to consider the function when \(x>2\). $$ \lim_{x \rightarrow 2^{+}} f(x) = \lim_{x \rightarrow 2^{+}} (x-2) $$ Again, the function is continuous for \(x > 2\), so we can evaluate the limit directly by substitution: $$ \lim_{x \rightarrow 2^{+}} f(x) = 2-2 = 0 $$

Step 3: Set the left limit and right limit equal to each other

Now that we've found both the left limit and the right limit, we can set them equal to each other, since the limit exists only if these two limits are equal. $$ 6+b = 0 $$

Step 4: Solve for b

To find the value of b for which the limit exists, we simply have to solve for b in the above equation: $$ b = -6 $$ With b=-6, \(\lim_{x \rightarrow 2} f(x)\) will exist.

Step 5: State the value of the limit

Since we know that the left and right limits are equal when \(b=-6\), we can state the value of the limit when \(x \rightarrow 2\): $$ \lim_{x \rightarrow 2} f(x) = \lim_{x \rightarrow 2^{+}} f(x) = 0 $$ So, the value of the limit as \(x\rightarrow 2\) is 0, if \(b=-6\).

Key Concepts

Left-Hand Limit

The concept of a left-hand limit in calculus is crucial when we analyze the behavior of a function as the input approaches a certain point from the left side. Specifically, when we talk about the limit of a function like f(x) as x approaches a value a from the left, we denote it as \(lim_{{x \rightarrow a^{-}}} f(x)\). What this means in practice is that we are looking at the values of f(x) that get closer and closer to x = a, without actually reaching a, and only from values less than a.

In our problem, to find the left-hand limit of f(x) as x approaches 2, we only consider the piece of the function where x \(\leq\) 2. The direct substitution method works here since the piecewise function is continuous when approaching from the left. Hence, the left-hand limit is 6 + b.

Right-Hand Limit

Conversely, the right-hand limit involves approaching the value a from the right side. It is written as \(\lim_{{x \rightarrow a^{+}}} f(x)\) and focuses on the values of f(x) where x is greater than a but getting infinitesimally close to it. This one-sided limit helps in understanding the behavior of functions that may not be continuous or may behave differently as the variable approaches from different sides.

In the context of our exercise, the right-hand limit of f(x) as x approaches 2 requires us to look at the function definition for x>2. The piecewise nature of the function demands that we consider the expression x-2 in this case. By substituting x with 2, we find that the right-hand limit is 0.

Piecewise Functions

Piecewise functions are defined by different expressions depending on the interval of the input value x. They can demonstrate a variety of behaviors, as different rules apply to different sections of their domains. When working with piecewise functions, it's critical to carefully consider the conditions under which each piece of the function applies.

In our particular example, the function f(x) is given by one rule, 3x + b, when x \(\leq\) 2, and by another rule, x - 2, when x > 2. To solve problems involving piecewise functions, we must evaluate each piece independently while considering the transition points where the function's definition changes.

Continuity in Calculus

Continuity is a fundamental concept in calculus that describes a function that is unbroken or smooth over its domain. A continuous function has no gaps, jumps, or points of disconnection. For a function f(x) to be continuous at a point x = a, three conditions must be met: the function must be defined at a, the limit as x approaches a must exist, and the limit must equal the function's value at a.

The step by step solution seamlessly showcases this concept when it finds a value of b for which \(\lim_{{x \rightarrow 2}} f(x)\) exists. With the correct value of b found to be -6, the left and right limits align with each other implying the function's continuity at that point. If a piecewise function has consistent limits from both sides at a critical point and equals the function’s value at that point, it is considered continuous there.