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Chapter 1: Problem 66

Compute the following limits. \(\lim _{x \rightarrow \infty} \frac{x^{2}+x}{x^{2}-1}\)

Short Answer

Expert verified
The limit is 1.

Step by step solution

01

Identify the type of limit

The limit given is \(\frac{x^{2}+x}{x^{2}-1}\), as \( x \rightarrow \infty \). Recognize that both the numerator and the denominator are polynomials of degree 2.
02

Rewrite the limit

Rewrite the limit expression by dividing every term by the highest power of \( x \) in the denominator (which is \( x^2 \)). This gives: \[ \frac{x^{2}/x^{2} + x/x^{2}}{x^{2}/x^{2} - 1/x^{2}} = \frac{1 + 1/x}{1 - 1/x^{2}} \]
03

Apply the limit

Now apply the limit as \( x \rightarrow \infty \) to the simplified expression. As \( x \rightarrow \infty \), the terms \( 1/x \) and \( 1/x^{2} \) approach 0. Thus, the expression becomes: \[ \frac{1 + 0}{1 - 0} = 1 \]
04

Conclusion

The limit of the given function as \( x \rightarrow \infty \) is 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomials
Polynomials are mathematical expressions consisting of variables raised to positive integer powers and multiplied by coefficients. In the limit problem given, both the numerator and the denominator are polynomial expressions of degree 2. The numerator is \(x^2 + x\), and the denominator is \(x^2 - 1\). These kinds of expressions are common in calculus problems, particularly when dealing with limits at infinity.
Infinity
Infinity, often denoted by the symbol \( \infty \), plays a crucial role in calculus, especially in limits. When a variable approaches infinity, it grows without bound. In our exercise, we analyze what happens to the function \(\frac{x^2 + x}{x^2 - 1}\) as \(x\) increases indefinitely. This is done by simplifying the expression and observing the behavior of each term. As \(x\) tends to infinity, certain terms like \(\frac{1}{x}\) and \(\frac{1}{x^2}\) approach zero, simplifying the computations.
Simplification Techniques
Simplification techniques help transform complex mathematical expressions into simpler forms, making them easier to evaluate. In the problem at hand, the expression \(\frac{x^2 + x}{x^2 - 1}\) is simplified by dividing each term by the highest power of \(x\) in the denominator, which is \(x^2\). Here鈥檚 the step-by-step breakdown:
  • Rewrite the terms: \(\frac{x^2}{x^2} + \frac{x}{x^2} \) and \(\frac{x^2}{x^2} - \frac{1}{x^2}\).
  • This simplifies to \(\frac{1 + \frac{1}{x}}{1 - \frac{1}{x^2}}\).

As \(x\) approaches infinity, both \(\frac{1}{x}\) and \(\frac{1}{x^2}\) approach zero. Thus, the expression further reduces to \(\frac{1 + 0}{1 - 0}\), which equals 1. These simplification techniques are fundamental in solving limit problems efficiently.

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Most popular questions from this chapter

Use limits to compute \(f^{\prime}(x)\). [Hint: In Exercises \(45-48\), use the rationalization trick of Example \(8 .]\) \(f(x)=\frac{x}{x+2}\)

The position of a particle moving on a line is given by \(s(t)-2 t^{3}-21 t^{2}+60 t, t \geq 0\), where \(t\) is measured in seconds and \(s\) in feet. (a) What is the velocity after 3 seconds and after 6 seconds? (b) When is the particle moving in the positive direction? (c) Find the total distance traveled by the particle during the first 7 seconds.

Compute the following limits. \(\lim _{x \rightarrow-\infty} \frac{1}{x^{2}}\)

A toy company introduces a new video game on the market. Let \(S(x)\) denote the number of videos sold on the day \(x\) since the item was introduced. Let \(n\) be a positive integer. Interpret \(S(n), S^{\prime}(n)\), and \(S(n)+S^{\prime}(n)\)

Use limits to compute \(f^{\prime}(x)\). [Hint: In Exercises \(45-48\), use the rationalization trick of Example \(8 .]\) \(f(x)=\frac{x}{x+1}\)
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