Problem 15 - Precalculus: Mathematics for Calculus

Question

Find the limit. $$\lim _{x \rightarrow-\infty}\left(\frac{x-1}{x+1}+6\right)$$

Step-by-Step Solution

Answer

The limit is 7.

Step 1: Identify the Limit Problem

The expression we are considering is \( \lim_{x \to -\infty} \left( \frac{x-1}{x+1} + 6 \right) \), which involves finding the behavior of the function as \( x \to -\infty \).

Step 2: Simplify the Fraction

Consider the fraction \( \frac{x-1}{x+1} \). Dividing the numerator and the denominator by \( x \) gives: \( \frac{1 - \frac{1}{x}}{1 + \frac{1}{x}} \). As \( x \to -\infty \), the terms \( \frac{1}{x} \) approach 0.

Step 3: Evaluate the Limit of the Fraction

The limit of \( \frac{1 - \frac{1}{x}}{1 + \frac{1}{x}} \) as \( x \to -\infty \) is \( \frac{1 - 0}{1 + 0} = 1 \). Thus, \( \lim_{x \to -\infty} \frac{x-1}{x+1} = 1 \).

Step 4: Combine with the Constant

Now, add the constant 6 to the limit of the fraction. We have: \( 1 + 6 = 7 \).

Step 5: Final Answer

Thus, \( \lim_{x \to -\infty} \left( \frac{x-1}{x+1} + 6 \right) = 7 \).

Key Concepts

Asymptotic Behavior

When you hear the term "asymptotic behavior," it refers to how a function behaves as it approaches a particular point or infinity. In other words, it's the direction the function is heading toward as the variable gets very large (positive or negative) or approaches a specific value. For instance, understanding the asymptotic behavior helps us know if a function gets closer and closer to a horizontal line, known as an asymptote, as the input values get extremely large or small. This is particularly relevant when examining the limit of rational functions at infinity, like in our problem where we look at how the given rational expression behaves as \( x \rightarrow -\infty \).

In our specific limit problem, we noticed how \( \frac{x-1}{x+1} \) approaches a constant as \( x \) becomes very large in the negative direction. This constant behavior is an aspect of asymptotic behavior, illustrating how despite the complexity of a function, at infinity it might resemble a simple line or a constant.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. In our exercise, we worked with the rational function \( \frac{x-1}{x+1} \). These types of functions often have interesting behavior as \( x \rightarrow \pm \infty \), making it easy to analyze their limits and asymptotic behaviors.

With rational functions, one helpful method to understand their limits is to simplify by dividing both the numerator and denominator by the highest power of \( x \) found in the denominator. Doing this can help you clearly see how terms behave as \( x \) approaches infinity.

• For example, in \( \frac{x-1}{x+1} \), dividing both parts by \( x \) gives \( \frac{1 - \frac{1}{x}}{1 + \frac{1}{x}} \).
• When \( x \) becomes very large, the terms containing \( \frac{1}{x} \) tend toward zero.

Recognizing these transformations helps us determine that rational functions can often approach a horizontal line, making them easier to work with when finding limits.

Infinity in Limits

Infinity in limits is a way to express what happens to a function as the variable either increases or decreases without bound. It's not an actual number, but more of a concept showing indefinite growth or shrinkage. When we're dealing with limits at infinity, such as \( \lim_{x \to -\infty} \), we’re observing how a function stretches out its behavior as \( x \) travels toward either positive or negative infinity.

In our problem, when we calculate \( \lim_{x \to -\infty} \left( \frac{x-1}{x+1} + 6 \right) \), it reminds us that we can still find real numbers representing how the function behaves at extreme values of \( x \). Finding these limits helps us see the broader shape or trend of the graph of a function without having to plot lots of points. Particularly when we have rational functions, determining limits at infinity showcases how even unwieldy expressions can simplify and align with a simple numerical trend as \( x \) grows very large positively or negatively. Understanding this core idea allows us to effectively predict the long-term trends of various functions.