Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. $$ \lim _{x \rightarrow \infty} \sqrt{\frac{x^{2}+x+3}{(x-1)(x+1)}} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Analyze the Limit Expression

We are given the limit \( \lim _{x \rightarrow \infty} \sqrt{\frac{x^{2}+x+3}{(x-1)(x+1)}} \). We know that evaluating limits as \( x \to \infty \) typically involves simplifying the expression by eliminating lower order terms.

Step 2: Simplify the Expression Before Taking the Limit

We start simplifying \( \frac{x^2 + x + 3}{(x-1)(x+1)} \). First, note that:\[ (x-1)(x+1) = x^2 - 1 \]Thus, the expression becomes:\[ \frac{x^2 + x + 3}{x^2 - 1} \]

Step 3: Simplify the Rational Expression

Divide the numerator and the denominator by \( x^2 \), the highest power of \( x \):\[ \frac{1 + \frac{1}{x} + \frac{3}{x^2}}{1 - \frac{1}{x^2}} \].As \( x \to \infty \), the terms \( \frac{1}{x} \) and \( \frac{1}{x^2} \) approach zero.

Step 4: Take the Limit of the Simplified Expression

Now, evaluate the limit:\[ \frac{1 + 0 + 0}{1 - 0} = 1 \].So, the limit simplifies to 1.

Step 5: Find the Square Root

Since we have the limit inside the square root, calculate:\[ \sqrt{1} = 1 \].

Key Concepts

Infinite Limits

When dealing with calculus, encountering limits where the variable approaches infinity can initially seem perplexing. However, the basic idea is to understand how functions behave when their input grows extremely large. In the exercise, you examine the expression as \( x \) moves towards infinity.

The key strategy for evaluating these infinite limits involves focusing on the terms with the highest power of \( x \). Higher power terms dominate the behavior of a function as \( x \) increases. Therefore, when faced with such limits, we aim to simplify expressions to reveal these dominant terms.

This approach effectively helps us ignore smaller terms which diminish in influence relative to the dominating terms. This simplification lies at the heart of finding infinite limits.

Rational Expressions

A rational expression is a fraction in which both the numerator and the denominator are polynomials. In the given exercise, \( \frac{x^2 + x + 3}{(x-1)(x+1)} \) exemplifies a rational expression. Such expressions often arise within calculus problems, particularly in limit problems. Simplifying these expressions is crucial before taking limits, especially when approaching infinity.

To simplify rational expressions, like the one on hand, we usually perform operations such as factoring and canceling terms. In this exercise, you begin by recognizing that the denominator \((x-1)(x+1)\) expands to \(x^2 - 1\). This simplifies the rational expression into a form that is easier to handle.

The simplification process illuminates the structure of the rational expression, making it straightforward to handle infinite limits. This allows for easier manipulation on the path towards finding the limit.

Limit Simplification

Limit simplification is a critical step in effectively evaluating limits of complex expressions. This step involves reframing the expression by removing negligible or canceling terms, making the evaluation straightforward.

For instance, in the problem, the task is to analyze \( \sqrt{\frac{x^2 + x + 3}{x^2 - 1}} \) as \( x \) approaches infinity. We begin by dividing both the numerator and the denominator by \( x^2 \), the highest degree term. The aim is to simplify the expression to \( \frac{1 + \frac{1}{x} + \frac{3}{x^2}}{1 - \frac{1}{x^2}} \).

As \( x \) grows infinitely large, terms like \( \frac{1}{x} \) and \( \frac{1}{x^2} \) approach zero. This renders the expression simpler: \( \frac{1 + 0 + 0}{1 - 0} = 1 \). Applying the square root gives you the final answer: 1. Limit simplification demystifies the problems, letting you resolve what might initially appear complex.