Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

, find the indicated limit. In most cases, it will be wise to do some algebra first. $$ \lim _{t \rightarrow 7^{+}} \frac{\sqrt{(t-7)^{3}}}{t-7} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Analyze the Problem

We need to find the limit of the expression \( \frac{\sqrt{(t-7)^3}}{t-7} \) as \( t \) approaches 7 from the right. Simply substituting \( t = 7 \) leads to an indeterminate form (\( \frac{0}{0} \)). Thus, we should try algebraic manipulation.

Step 2: Simplify the Expression

Recognizing \( \sqrt{(t-7)^3} = (t-7)^{3/2} \), we can rewrite the original expression as \( \frac{(t-7)^{3/2}}{t-7} \). This allows us to simplify to \((t-7)^{1/2} \) by cancelling out \( t-7 \) from the numerator and denominator.

Step 3: Calculate the Limit

Now, calculate the limit of the simplified expression \((t-7)^{1/2}\) as \( t \to 7^{+} \). Substituting \( t = 7 \), we have \((7-7)^{1/2} = 0^{1/2} = 0 \). Thus, the limit is 0.

Key Concepts

Indeterminate Forms

When calculating limits, you might encounter expressions that at first glance seem challenging to evaluate. These are known as **indeterminate forms**. One common example is the form \( \frac{0}{0} \). This occurs when both the numerator and denominator of a fraction approach zero. In situations like these, substituting the value directly does not yield a result. Instead, the expression is undefined, and we must use another method to proceed.
Some other types of indeterminate forms include \( \frac{\infty}{\infty} \), \( 0\times\infty \), and \( \infty - \infty \). When faced with one of these, it's crucial to manipulate the expression to find a limit that is determinate. Special techniques, such as algebraic manipulation or L'Hôpital's Rule, can help us resolve these issues and find the actual limit.

Algebraic Manipulation

**Algebraic manipulation** plays a vital role in solving limit problems, especially when dealing with indeterminate forms. By rewriting expressions, we can often eliminate the troublesome parts that cause the indeterminate form.
In the given exercise, we have the expression \( \frac{\sqrt{(t-7)^3}}{t-7} \). Initially, substituting \( t = 7 \) results in a \( \frac{0}{0} \) form. To work around this, notice that \( \sqrt{(t-7)^3} \) can be rewritten as \((t-7)^{3/2}\). By expressing the denominator as \((t-7)^1\), we can simplify the fraction to \((t-7)^{1/2}\).
This simplification removes the \( t-7 \) that was causing the division by zero and allows us to directly compute the limit. By using basic algebraic techniques, such as factoring and cancelling, we often find clearer paths to solve problems.

Simplifying Expressions

After algebraic manipulation, **simplifying expressions** is the next step in making your calculations much more straightforward. By reducing the expression to its simplest form, you open the door to easily evaluating limits.
In our problem, simplifying \( \frac{(t-7)^{3/2}}{t-7} \) to \((t-7)^{1/2}\) made it clear and computable. The \( t-7 \) terms effectively canceled each other out, and you were left with an expression that was no longer problematic.
Simplifying does not just make the expression neater; it often transforms it into a form that directly shows the behavior of the function as it approaches a specific point. This simplification is essential for understanding how functions behave and for finding limits effectively. Remember that simplification is a tool of clarity, aligning complex algebra with easier interpretations.