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Chapter 2: Problem 26

Find the limits. $$\lim _{x \rightarrow+\infty} \frac{\sqrt{3 x^{4}+x}}{x^{2}-8}$$

Short Answer

Expert verified
The limit is \(\sqrt{3}\).

Step by step solution

01

Identify Dominant Terms

To find the limit as \(x\) approaches infinity, first identify the dominant terms in the numerator and the denominator. For \(\sqrt{3x^4 + x}\), the dominant term is \(\sqrt{3x^4}\) as \(x\to \infty\). Similarly, in the denominator \(x^2 - 8\), the dominant term is \(x^2\).
02

Simplify the Expression

Rewrite the expression using the dominant terms: \(\lim_{x\to\infty} \frac{\sqrt{3x^4+x}}{x^2-8} \approx \lim_{x\to\infty} \frac{\sqrt{3x^4}}{x^2}\). Simplify \(\frac{\sqrt{3x^4}}{x^2}\) to \(\frac{\sqrt{3}x^2}{x^2} = \sqrt{3}\).
03

Apply the Limit

Since both the numerator and the denominator have been dominated by the terms \(\sqrt{3}x^2\) and \(x^2\), respectively, the expression reduces to \(\sqrt{3}\) as \(x\to\infty\). Thus, the limit is \(\sqrt{3}\).

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Key Concepts

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

Dominant Terms
When dealing with limits in calculus, especially as variables approach infinity, identifying dominant terms becomes crucial.
  • Dominant terms are the terms that have the most significant effect on the value of an expression as a variable grows very large.
  • In our example, the dominant term in the numerator \( \sqrt{3x^4 + x} \) is \( \sqrt{3x^4} \) because it grows much faster than \( x \) as \( x \to \infty \).
  • Similarly, for the denominator \( x^2 - 8 \), the term \( x^2 \) is dominant since it overpowers the constant \(-8\) as \( x \) becomes infinitely large.
Understanding which terms dominate lets us simplify complex expressions efficiently. Prioritize these terms when evaluating limits involving infinity, to find quick and accurate results.
Infinity in Calculus
Infinity is a fundamental concept in calculus that helps us understand behavior at extreme values.
  • With infinities, we are often interested in how functions grow compared to each other.
  • The expression shows the relationship between the numerator and denominator as both tend towards infinity.
  • By focusing on the dominant terms, we examine the highest growth rates. This helps us determine what the function approaches, if anything, as the variable becomes infinitely large or small.
These insights help in defining limit behavior, which is a cornerstone of calculus understanding, particularly when dealing with rational functions and their growth comparison.
Simplifying Expressions
Simplifying expressions is a powerful tool in calculus for handling complex limits. It involves rewriting an expression while keeping its value equivalent. In our scenario:
  • First, approximate using dominant terms: \( \frac{\sqrt{3x^4}}{x^2} \) simplifies from the original \( \frac{\sqrt{3x^4+x}}{x^2-8} \).
  • Next, simplify: the expression \( \frac{\sqrt{3x^4}}{x^2} \) becomes \( \frac{\sqrt{3}x^2}{x^2} \).
  • Cancel the common \( x^2 \), resulting in \( \sqrt{3} \), as both the numerator and denominator had a term proportional in growth.
Simplifying before computing the limit avoids dealing with unwieldy calculations in infinity. It highlights the constants and reduces errors, providing a clearer path to the solution.

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