Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

State whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ f(t)=\left\\{\begin{array}{ll} t-3 & \text { if } t \leq 3 \\ 3-t & \text { if } t>3 \end{array}\right. $$

Step-by-Step Solution

Answer

The function is continuous at \( t = 3 \) because both limits and the function value are 0.

Step 1: Determine the function behavior at given conditions

The function is piece-wise defined, meaning its behavior changes based on the value of \( t \). We must consider both cases: \( t \leq 3 \) and \( t > 3 \).

Step 2: Evaluate the left-hand limit as \( t \rightarrow 3^- \)

For \( t \leq 3 \), the function is defined as \( f(t) = t-3 \). As \( t \) approaches 3 from the left, \( f(t) = 3 - 3 = 0 \). Thus, the left-hand limit as \( t \rightarrow 3^- \) is 0.

Step 3: Evaluate the right-hand limit as \( t \rightarrow 3^+ \)

For \( t > 3 \), the function is defined as \( f(t) = 3-t \). As \( t \) approaches 3 from the right, \( f(t) = 3 - 3 = 0 \). Thus, the right-hand limit as \( t \rightarrow 3^+ \) is 0.

Step 4: Find the function value at \( t = 3 \)

The function for \( t = 3 \) is evaluated using the case where \( t \leq 3 \). Therefore, \( f(3) = 3 - 3 = 0 \).

Step 5: Check continuity condition at \( t = 3 \)

A function is continuous at a point if the left-hand limit, right-hand limit, and the function value at that point are all equal. Here, the left-hand limit, right-hand limit, and \( f(3) \) are all 0. Hence, \( f(t) \) is continuous at \( t = 3 \).

Key Concepts

Piecewise Function

In mathematics, a piecewise function is a type of function defined by different expressions over different intervals. It's like having a function that can change its rule depending on the value of the input. The idea is to divide the domain into intervals and assign a different expression to each interval. This allows the function to "switch" its behavior at specific points, making it very flexible for modeling real-world situations.

In the exercise, the function \( f(t) \) is defined as:

• \( t - 3 \) for \( t \leq 3 \)
• \( 3 - t \) for \( t > 3 \)

This means for values of \( t \) up to and including 3, we use \( t - 3 \), and for values of \( t \) greater than 3, we switch to \( 3 - t \). Understanding piecewise functions is crucial because the behavior of these functions can significantly differ across different parts of their domains.

Limits

The concept of limits is fundamental in calculus and helps us understand the behavior of a function as the input approaches a certain point. Limits give us an idea of what value a function 'wants to be' when nearing a specific input value, even though it might not actually reach that value.

Imagine a car approaching a stop sign. Although the car may not yet be fully stopped, we can predict its stopping point based on its movement. Similarly, for mathematical functions, limits allow us to analyze how the function behaves towards a boundary or significant point, which is essential when dealing with piecewise functions that may seem discontinuous at first glance due to their change in rules.

Left-hand Limit

The left-hand limit is the value that a function approaches as its input approaches a given point from the left direction. Essentially, it captures the behavior of a function just before it reaches a specific value.

For the given function \( f(t) \), we evaluate the left-hand limit as \( t \to 3^- \). Here, we use the expression \( t - 3 \) because \( t \leq 3 \). Plugging \( t = 3 \) into \( f(t) \), the calculation becomes \( 3 - 3 = 0 \). Therefore, the left-hand limit of the function at \( t = 3 \) is 0.

Understanding left-hand limits is important because they are part of the continuity check, ensuring that the function behaves predictably as we approach certain points from the left.

Right-hand Limit

The right-hand limit is the value a function approaches as the input comes from the right side of a given point. It describes what the function is trending towards immediately after reaching a certain value.

In our example, to find the right-hand limit as \( t \rightarrow 3^+ \), we use the expression \( 3 - t \) because it is defined for \( t > 3 \). When we substitute \( t = 3 \), we calculate \( 3 - 3 = 0 \). Thus, the right-hand limit at \( t = 3 \) is also 0.

Knowing the right-hand limit is crucial for assessing the continuity of a function, making sure that the function's behavior matches expectations when transitioning from certain points to slightly larger ones. Together with the left-hand limit and function value, this verifies whether or not a function is continuous at a point.