Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the limits. A graph may be useful. \(f(x)=\frac{|x-3|}{x-3}\) (a) \(\lim _{x \rightarrow 0} f(x)\) (b) \(\lim _{x \rightarrow 4} f(x)\) (c) \(\lim _{x \rightarrow 3^{+}} f(x)\) (d) \(\lim _{x \rightarrow 3^{-}} f(x)\) (c) \(\lim _{x \rightarrow 3} f(x)\)

Step-by-Step Solution

Answer

The volumes of the limits are \(\lim _{x \rightarrow 0} f(x) = -1\), \(\lim _{x \rightarrow 4} f(x) = 1\), \(\lim _{x \rightarrow 3^{+}} f(x) = 1\), \(\lim _{x \rightarrow 3^{-}} f(x) = -1\), and \(\lim _{x \rightarrow 3} f(x)\) doesn't exist.

Step 1: Find the limit as x approaches from the left of 3

Since \(x < 3\), the absolute value function \(|x-3|\) will return \(-(x-3)\). So, \(f(x)=\frac{|x-3|}{x-3}\) can be rewritten as \(f(x)=\frac{-(x-3)}{x-3} = -1\). Hence, \(\lim _{x \rightarrow 3^{-}} f(x)\) is -1.

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

Since \(x > 3\), the absolute value function \(|x-3|\) will return \(x-3\). So, \(f(x)=\frac{|x-3|}{x-3}\) can be rewritten as \(f(x)=\frac{x-3}{x-3} = 1\). Hence, \(\lim _{x \rightarrow 3^{+}} f(x)\) is 1.

Step 3: Find the limit as x approaches from 0

As per the definition of the function, when we plug in the value \(x=0\) into \(f(x)\), the expression becomes \(f(0)=\frac{|0-3|}{0-3}\), which evaluates to -1. Therefore, \(\lim _{x \rightarrow 0} f(x) = -1\).

Step 4: Find the limit as x approaches from 4

When we plug in the value \(x=4\) into \(f(x)\), the expression becomes \(f(4)=\frac{|4-3|}{4-3}\), which evaluates to 1. Therefore, \(\lim _{x \rightarrow 4} f(x) = 1\).

Step 5: Find the limit as x approaches 3

Since the limit of the function as x approaches 3 from the left doesn't equal to the limit as x approaches from the right, the overall limit at x=3 does not exist. Therefore, \(\lim _{x \rightarrow 3} f(x)\) doesn't exist.

Key Concepts

Limits of Piecewise Functions

Understanding the limits of piecewise functions involves grasping how a function behaves differently in distinct intervals. With piecewise functions, such as the absolute value function in our example, the limit can differ based on the input value's proximity to the point of interest from either side.

For instance, our function \(f(x)=\frac{|x-3|}{x-3}\) behaves differently to the left and right of \(x=3\), effectively splitting the analysis into two separate cases. When approaching the limit from the left (\(x < 3\)), we consider the negative part of the function; when approaching from the right (\(x > 3\)), we consider the positive part.

Consequently, the limit as \(x\) approaches 3 from either side will yield different results, and as such, if the function does not converge to the same value from both directions, the limit at that point does not exist. It's essential for students to consider each interval on its own to correctly evaluate these limits.

Absolute Value Functions

Absolute value functions are unique in that they return the non-negative magnitude of a number or expression, regardless of its original sign. In our example, \(f(x)=\frac{|x-3|}{x-3}\), the absolute value of \(x-3\) is crucial in evaluating the limit.

For \(x\) values less than 3, \(x-3\) is negative, thus \(|x-3|\) is its positive counterpart, resulting in a negative expression when divided by \(x-3\). For \(x\) values greater than 3, \(|x-3|\) returns \(x-3\), yielding a positive result when the expression is divided by \(x-3\).

This switch in sign is pivotal and alters the outcome of the limit based on whether we approach from the left or right side of the specified value. Understanding the behavior of absolute value expressions within limits is a fundamental step in correctly solving these problems.

One-Sided Limits

One-sided limits are a way to describe the behavior of a function as the input approaches a specific value from one side only, either from the left (\(x \rightarrow c^{-}\)) or from the right (\(x \rightarrow c^{+}\)). These limits are critical when a function has a sudden change or discontinuity at the value \(c\).

In our example, we explore both one-sided limits as \(x\) approaches 3. When calculating \(\lim _{x \rightarrow 3^{-}} f(x)\), we're interested in the behavior of \(f(x)\) as it approaches 3 from the left. Conversely, when calculating \(\lim _{x \rightarrow 3^{+}} f(x)\), we look at its behavior from the right.

The results of these calculations inform us whether or not the function is continuous around \(x=3\) or if there's a jump or break at that point. The distinction between left and right-hand limits aids in understanding the overall behavior of the function, highlighting possible points of discontinuity.

Limits and Continuity

Limits and continuity are deeply interconnected concepts within calculus. A function is continuous at a point if the left-hand limit, right-hand limit, and the function's value at that point all exist and are equal. Any discrepancy in these values suggests a discontinuity.

Applying this understanding to our function \(f(x)\), we determine that since \(\lim _{x \rightarrow 3^{-}} f(x) eq \lim _{x \rightarrow 3^{+}} f(x)\), the function is not continuous at \(x=3\). The limit does not exist here because the behavior of \(f(x)\) as it approaches from the left is not the same as when it approaches from the right.

This concept highlights that the evaluation of a limit is intrinsic to the concept of continuity. It's important for students to not just calculate limits mechanically but to use them as a tool for understanding the broader behavior of functions, particularly at points where the function may not behave 'nicely' or intuitively.