Problem 26 - Calculus: Early Transcendentals

Question

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. \( \displaystyle \lim_{t \to 0}\frac{5^t - 1}{t} \)

Step-by-Step Solution

Answer

The estimated limit is approximately 1.48.

Step 1: Understanding the Limit Expression

We need to estimate the limit of the expression \( \frac{5^t - 1}{t} \) as \( t \) approaches 0. This means that we should evaluate how the expression behaves as \( t \) gets closer and closer to zero from both the positive and negative sides.

Step 2: Create a Table of Values

To create a table of values, we'll select values of \( t \) that are close to 0, such as \( t = -0.1, -0.01, -0.001, 0, 0.001, 0.01, 0.1 \). For each of these values, we will compute \( \frac{5^t - 1}{t} \).

Step 3: Calculate Values for Positive \( t \) Near 0

Let's calculate \( \frac{5^t - 1}{t} \) for positive values: - For \( t = 0.1 \), \( \frac{5^{0.1} - 1}{0.1} \approx 1.61 \)- For \( t = 0.01 \), \( \frac{5^{0.01} - 1}{0.01} \approx 1.48 \)- For \( t = 0.001 \), \( \frac{5^{0.001} - 1}{0.001} \approx 1.45 \)

Step 4: Calculate Values for Negative \( t \) Near 0

Now, calculate for negative values of \( t \): - For \( t = -0.001 \), \( \frac{5^{-0.001} - 1}{-0.001} \approx 1.45 \)- For \( t = -0.01 \), \( \frac{5^{-0.01} - 1}{-0.01} \approx 1.48 \)- For \( t = -0.1 \), \( \frac{5^{-0.1} - 1}{-0.1} \approx 1.61 \)

Step 5: Analyze and Estimate the Limit

Observe that as \( t \) approaches 0 from both the positive and negative directions, the value of \( \frac{5^t - 1}{t} \) seems to converge towards 1.48. Therefore, we estimate that the limit is approximately 1.48.

Step 6: Verify using a Graphing Calculator

Plot the function \( f(t) = \frac{5^t - 1}{t} \) on a graphing calculator. The graph should show that as \( t \) approaches 0, the y-values of the graph approach 1.48. This graphical verification confirms our estimated limit.

Key Concepts

Limit Estimation

Limit estimation is the process of finding the value that a function approaches as its input approaches a certain value. In our example, we need to estimate the limit of the expression \( \frac{5^t - 1}{t} \) as \( t \) approaches 0. This is particularly interesting because if we try to substitute \( t = 0 \) directly into the expression, we get an undefined form \( \frac{0}{0} \). Therefore, we need other methods such as numerical and graphical approaches.

One popular way to estimate limits is by choosing values of \( t \) that are very close to 0 and examining how the expression behaves. This involves checking the expression's value from both sides of the point we are interested in, paying close attention to whether these values converge towards a single number as we approach the point. This single number is what we refer to as the estimated limit of the function.

Graphical Verification

Graphical verification involves using visual tools such as graphs to confirm the results obtained through mathematical estimations. For our problem, we plot the function \( f(t) = \frac{5^t - 1}{t} \) on a graphing calculator. As we focus on how the graph behaves when \( t \) approaches 0, the visual representation provides insights into how the function behaves near this point.

The graph is expected to show a clear trend of the y-values approaching a certain number as \( t \) gets closer to 0. In our specific example, we predict from our calculations that the function approaches the limit of approximately 1.48. If the graph confirms this by showing the curve approaching y = 1.48 as \( t \) moves towards zero from both sides, then the graphical verification supports our estimated limit.

Table of Values

Creating a table of values is a practical method to explore how a function behaves near the point of interest. For our example, we select values of \( t \) that are very close to 0, such as \( -0.1, -0.01, -0.001, 0, 0.001, 0.01, 0.1 \), and compute \( \frac{5^t - 1}{t} \) for each.

Listing these results helps us to see patterns. Specifically:

• For positive values of \( t \), it appears the values decrease towards a point.
• Similarly, for negative \( t \), the values increase towards that same point.

Both trends nearing a value of approximately 1.48 as \( t \) approaches zero, thus pointing out roughly what the limit should be. The table acts as concrete evidence in our estimation process, offering a numerical look into the limit behavior.

Exponential Functions

Exponential functions, like \( 5^t \) in our limit problem, exhibit rapid growth or decay depending on the exponent \( t \). When working with limits involving such functions, the key trait is how they behave with small values of \( t \).

For very small \( t \), \( 5^t \) is close to 1, which indicates a small change or deviation about the base value. In our case, since we compute \( \frac{5^t - 1}{t} \), the expression accounts for how \( 5^t \) differs from 1, adjusted by \( t \), leading us to explore the instantaneous rate of change at \( t = 0 \). This links directly to the concept of derivatives in calculus, where analyzing how the output of an exponential function changes as \( t \) gets extremely close to zero is crucial.

Such nuances highlight the powerful nature of exponential functions in limit problems, demanding attention to subtle variances around the pivoting point, which in this case, is 0.