Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

State whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ r(t)=\left\\{\begin{array}{ll} \frac{t^{3}-27}{t-3} & \text { if } t \neq 3 \\ 23 & \text { if } t=3 \end{array}\right. $$

Step-by-Step Solution

Answer

The function is not continuous at 3 because the limit and the function value differ.

Step 1: Define Continuity at a Point

A function \(f(x)\) is continuous at \(x = a\) if the following three conditions are met: 1. \(f(a)\) is defined. 2. The limit \(\lim_{x \to a} f(x)\) exists. 3. \(\lim_{x \to a} f(x) = f(a)\). We will apply these conditions to the function \(r(t)\) at \(t = 3\).

Step 2: Ensure Function Value at the Point

Verify because \(r(t) = 23\) for \(t = 3\), we have \(r(3) = 23\). Thus, the first condition of continuity is satisfied.

Step 3: Find the Limit from Both Sides

Calculate the limit of \(r(t)\) as \(t\) approaches 3. We need to simplify \(\frac{t^3 - 27}{t - 3}\) when \(t eq 3\). Notice that \(t^3 - 27\) can be factored using the difference of cubes: \((t - 3)(t^2 + 3t + 9)\). The expression simplifies to \(t^2 + 3t + 9\) when \(t eq 3\). Thus: \[\lim_{t \to 3} r(t) = \lim_{t \to 3} \left(t^2 + 3t + 9\right) = 9 + 9 + 9 = 27.\]

Step 4: Compare Limit and Function Value

Now, compare the limit found in step 3 to the function value at \(t = 3\). 1. \(\lim_{t \to 3} r(t) = 27\).2. However, \(r(3) = 23\). Since \(\lim_{t \to 3} r(t) eq r(3)\), the function is not continuous at \(t = 3\).

Key Concepts

Limits

Limits play a crucial role in understanding continuity. When we talk about limits, we focus on what happens to a function as the input approaches a particular value. In the context of this exercise, we want to know what the function \( r(t) \) does as \( t \) gets close to 3. This doesn't mean we are interested in what happens "at" 3, but rather what happens as we "approach" 3 from both sides.
To compute the limit \( \lim_{t \to 3} r(t) \) for the function given, we need to simplify the expression \( \frac{t^3 - 27}{t - 3} \) when \( t eq 3 \).
Notice that it factors to \((t - 3)(t^2 + 3t + 9)\). Canceling the \(t-3\) terms gives us \(t^2 + 3t + 9\), allowing us to directly substitute \(t = 3\) to find the limit, which results in 27. Thus, the limit value as \( t \to 3 \) is 27.
Understanding this kind of limit calculation is at the heart of investigating continuity at a point.

Function Evaluation

Function evaluation tells us what the actual value of the function is at a specific point. This is important because one of the conditions for continuity is that the function must be defined at the point of interest.
In our exercise, we need to evaluate \( r(t) \) at \( t = 3 \). According to the piecewise definition of the function, when \( t = 3 \), \( r(t) = 23 \). So, the function is certainly defined at \( t = 3 \). This ensures that the first condition for continuity, which is that \( f(a) \) is defined, is satisfied.
However, simply having the function defined at a specific point is not enough for continuity; this value also needs to match the limit value, as we will see in the analysis of discontinuity.

Discontinuity Analysis

Discontinuity occurs when the conditions for a function to be continuous aren't fully met. As reviewed earlier, there are three conditions for continuity:

• The function must be defined at the point.
• The limit must exist as the point is approached.
• The limit and the function value at that point must be equal.

While evaluating \( r(t) \) at \( t = 3 \), we saw that \( r(3) = 23 \). But we also determined that \( \lim_{t \to 3} r(t) = 27 \).
Here is where the function falters in terms of continuity. The mismatch between the limit and the actual function value — specifically, \( \lim_{t \to 3} r(t) eq r(3) \) — signals a discontinuity at \( t = 3 \).
Therefore, the function is not continuous at \( t = 3 \) because the limit at that point does not equal the actual value of the function. This type of discontinuity can often be addressed by redefining or modifying the function, but in this case, that is not part of our task.