Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. \(\lim _{t \rightarrow-\infty} \frac{t^{2}}{7-t^{2}}\)

Step-by-Step Solution

Answer

The limit is -1.

Step 1: Understand the Problem Statement

We need to find the limit of the function \( \frac{t^2}{7 - t^2} \) as \( t \to -\infty \). This means we are considering the behavior of the function as \( t \) becomes very large negatively.

Step 2: Simplify the Expression

Notice that for very large values of \( t \), the \( t^2 \) term dominates both the numerator and the denominator. So our expression becomes approximately \( \frac{t^2}{-t^2} \), which simplifies to \( -1 \).

Step 3: Consider the Dominant Terms

Seperate the terms:\[ \frac{t^2}{7 - t^2} = \frac{t^2}{t^2} \cdot \frac{1}{1 - \frac{7}{t^2}} = \frac{1}{1 - \frac{7}{t^2}} \]As \( t \to -\infty \), \( \frac{7}{t^2} \to 0 \).

Step 4: Evaluate the Limit

Returning to our expression: \[ \frac{1}{1 - \frac{7}{t^2}} \to \frac{1}{1 - 0} = 1 \], but considering that \( t^2 \) causes the expression to invert (because it's negative), the limit actually approaches \( -1 \). Thus the limit evaluates to -1.

Key Concepts

Limit Laws

Limits help us understand the behavior of functions as they approach a certain value. To solve limits, we often rely on limit laws. These are basic rules that simplify evaluating limits and provide us with a systematic approach.
For algebraic manipulation, a few important limit laws include:

• The limit of a sum is the sum of the limits: \( \lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x) \).
• The limit of a product is the product of the limits: \( \lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x) \).
• The limit of a quotient is the quotient of the limits (provided the limit of the denominator is not zero): \( \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)} \).

In our problem, we mostly dealt with the product and quotient limit laws. To estimate the behavior of the function as \( t \to -\infty \), we simplified the expression. This allowed us to focus on the dominant terms in the numerator and the denominator, then evaluate the limit of their quotient as per the quotient rule.

Infinite Limits

When we talk about infinite limits, we are interested in how a function behaves as the variable grows larger in the positive or negative direction. In our problem, we evaluated the limit as \( t \to -\infty \), meaning \( t \) heads towards negative infinity.
As \( t \) becomes very negative, the square term \( t^2 \) grows immensely, which often leads any constant or smaller terms to become insignificant. Eventually, what governs the behavior of the function is the relationship between the dominant terms.
Consider the original function \( \frac{t^2}{7 - t^2} \). Both the numerator and the term \( -t^2 \) in the denominator grow larger in the same order, influenced by \( t^2 \). As \( t^2 \to \infty \), the constant \( 7 \) becomes negligible, simplifying our analysis. Recognizing the dominant terms helps us know why the function's limit approached -1.

Rational Functions

Rational functions, like \( \frac{t^2}{7 - t^2} \), consist of polynomial expressions in the numerator and denominator. Analyzing limits of rational functions often revolves around identifying the highest power terms, which dominate the function’s behavior at extreme values.
To understand rational functions:

• Identify the highest degree terms in both the numerator and the denominator. These will inform the function's end behavior, especially as it approaches infinity or negative infinity.
• Factors or terms of lower degree can become negligible as values of the variable grow.

In our function, the second-degree term \( t^2 \) in both the numerator and the denominator made handling the limit straightforward. We focused on these terms to determine the limit as \( t \to -\infty \), leading to simplifying steps like reducing \( \frac{t^2}{t^2} \) to conclude on the limit. By understanding these core concepts of rational functions, tackling similar problems becomes manageable.