Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. \(\lim _{x \rightarrow \infty} \frac{x}{x-5}\)

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify the Function and Limit

We need to find the limit of the function \( f(x) = \frac{x}{x-5} \) as \( x \to \infty \). This means we want to understand the behavior of the function as \( x \) becomes very large.

Step 2: Simplify the Function

To find the limit, simplify the fraction \( \frac{x}{x-5} \) by dividing the numerator and the denominator by \( x \), the highest degree of \( x \) present in the expression. This gives you: \[ \frac{x}{x-5} = \frac{1}{1 - \frac{5}{x}} \]

Step 3: Evaluate the Limit

Now, evaluate the expression \( \frac{1}{1 - \frac{5}{x}} \) as \( x \to \infty \). As \( x \to \infty \), the term \( \frac{5}{x} \to 0 \), so the expression becomes:\[ \frac{1}{1 - 0} = \frac{1}{1} = 1 \]

Step 4: State the Limit

The limit of the function \( \frac{x}{x-5} \) as \( x \to \infty \) is 1. Thus, we have:\( \lim_{x \to \infty} \frac{x}{x-5} = 1 \)

Key Concepts

Infinite Limits

When exploring infinite limits, we're trying to determine what happens to a function as the input grows extremely large or very small. This concept is crucial for understanding the behavior at the extremes of a function. Infinite limits examine how a function behaves as the variable approaches infinity or negative infinity. Generally, we either find that the function approaches a finite number or it keeps growing or shrinking without bound.

Consider the function given in the exercise, \( \lim_{x \rightarrow \infty} \frac{x}{x-5} \). As \( x \to \infty \), we want to understand what value the function approaches. Analyzing the infinite limits helps us predict the eventual behavior and simplify the computation process. This simplification often involves observing dominant terms, like in our function, the linear term \( x \) in the numerator and denominator. Focusing on these helps us see the behavior more clearly and gives insight into whether the limit yields a finite result.

Rational Functions

Rational functions are ratios of two polynomials. They are frequently encountered in mathematics and present interesting limit problems, especially as variables approach infinity or other critical points. Understanding rational functions is central to making sense of how infinite limits work.

In our exercise, learning about the rational function \( \frac{x}{x-5} \) means observing how the structure produces an interesting behavior as \( x \to \infty \). Here, both the numerator and the denominator are linear polynomials. This symmetry often allows for simplification when calculating limits. In cases where polynomials have the same degree, the coefficients of the highest power terms usually determine the behavior as \( x \to \infty \). This is a critical property of rational functions worth understanding deeply.

Limit Simplification

Simplifying a limit is vital for making computations more tractable and connected with solving problems involving rational functions and infinite limits effectively. In our exercise, we simplified\( \frac{x}{x-5} \) by factoring out the highest degree term, namely, \( x \), from both the numerator and the denominator.

This simplification results in \( \frac{1}{1-\frac{5}{x}} \). This new expression offers clarity, as it plainly shows what happens as \( x \to \infty \). As \( x \to \infty \), the term \( \frac{5}{x} \to 0 \). This reveals the expression approaches \( \frac{1}{1} = 1 \). Simplification allows focus on the main contributors to the limit, ignoring the less impactful terms. This focus enables us to find limits quickly, particularly when analyzing rational functions.