Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

The given function is not defined at a certain point. How should it be defined in order to make it continuous at that point? $$ H(t)=\frac{\sqrt{t}-1}{t-1} $$

Step-by-Step Solution

Answer

Define \( H(1) = \frac{1}{2} \) to make it continuous at \( t = 1 \).

Step 1: Identify the Point of Discontinuity

The function is given as \( H(t) = \frac{\sqrt{t} - 1}{t - 1} \). The function is not defined at \( t = 1 \) because substituting \( t = 1 \) results in a division by zero.

Step 2: Use L'Hôpital's Rule

Since both the numerator and denominator approach zero as \( t \rightarrow 1 \), we apply L'Hôpital's Rule to find the limit as \( t \rightarrow 1 \). This involves differentiating the numerator and the denominator separately.

Step 3: Differentiate the Numerator and Denominator

The derivative of the numerator \( \sqrt{t} - 1 \) with respect to \( t \) is \( \frac{1}{2\sqrt{t}} \). The derivative of the denominator \( t - 1 \) is 1. Thus, the limit becomes \[ \lim_{t \to 1} \frac{\frac{1}{2\sqrt{t}}}{1}. \]

Step 4: Evaluate the Limit

Substitute \( t = 1 \) in the derivative expression: \( \frac{1}{2\sqrt{1}} = \frac{1}{2} \). Thus, the limit as \( t \to 1 \) of \( \frac{\sqrt{t} - 1}{t - 1} \) is \( \frac{1}{2} \).

Step 5: Define the Function at the Point

To make the function \( H(t) \) continuous at \( t = 1 \), we define \( H(1) = \frac{1}{2} \). This ensures that the limit of \( H(t) \) as \( t \) approaches 1 is equal to \( H(1) \).

Key Concepts

Discontinuity and Its Causes

In calculus, a function is said to be continuous at a point if its value at that point is well-defined and matches the limit of the function as it approaches that point. Discontinuity, on the other hand, happens when these conditions are not met. This often occurs in functions where there is an undefined point, typically due to division by zero or other such instances.
For the function \( H(t) = \frac{\sqrt{t} - 1}{t - 1} \), we encounter a discontinuity at \( t = 1 \). As we attempt to substitute \( t = 1 \) into the function, we end up dividing by zero, which renders the function undefined at that point.
To address this, we need to find how the function behaves around the discontinuity. Doing so allows us to redefine the function at that point if needed, aiming to remove the discontinuity.

Understanding and Applying L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus used to evaluate the limits of indeterminate forms such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). When a direct substitution into a limit results in these forms, L'Hôpital's Rule provides a way to resolve them by differentiating the numerator and the denominator individually.
In the case of the function \( H(t) = \frac{\sqrt{t} - 1}{t - 1} \) as \( t \to 1 \), we encounter the \( \frac{0}{0} \) form when substituting \( t = 1 \).
Applying L'Hôpital's Rule involves calculating the derivatives:

• The derivative of the numerator \( \sqrt{t} - 1 \) is \( \frac{1}{2\sqrt{t}} \).
• The derivative of the denominator \( t - 1 \) is \( 1 \).

Thus, \( \lim_{t \to 1} \frac{\sqrt{t} - 1}{t - 1} = \lim_{t \to 1} \frac{\frac{1}{2\sqrt{t}}}{1} = \frac{1}{2} \). This calculation reveals the behavior of the function as \( t \) approaches 1, allowing us to bypass the discontinuity.

The Role of Limits in Calculus

The concept of limits is foundational in calculus, as it formalizes how functions behave as they approach specific points or infinity. Limits essentially allow us to "peek" into the unobservable values of functions at critical points such as discontinuities or asymptotes.
In the given problem, calculating the limit of \( H(t) \) as \( t \) approaches 1 helps us understand the behavior of the function at the discontinuity. By using L'Hôpital's Rule and finding the limit to be \( \frac{1}{2} \), we can decide how to redefine the function:
- The computed limit suggests what value the function should take at the discontinuity point to be continuous.- Thus, by defining \( H(1) = \frac{1}{2} \), we can make the function nicely behave to ensure that it's continuous at \( t = 1 \).
Using limits in such contexts not only aids in resolving discontinuities but also plays a critical role in the analysis and understanding of the overall behavior of functions.