Problem 23 - Calculus Early Transcendentals

Question

15-36 Find the limit. $$ \lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{6}-x}}{x^{3}+1} $$

Step-by-Step Solution

Answer

The limit is 3.

Step 1: Identify the Dominant Terms

To solve the problem, identify the terms in both the numerator and the denominator that will dominate as \( x \to \infty \). In the numerator, the dominant term is \( \sqrt{9x^6} = 3x^3 \). In the denominator, the dominant term is \( x^3 \). This means these terms will primarily determine the behavior of the function as \( x \to \infty \).

Step 2: Simplify the Expression

Given the dominant terms, simplify the expression. We approximate: \( \sqrt{9x^6 - x} \approx \sqrt{9x^6} = 3x^3 \). Thus, the expression \( \frac{\sqrt{9x^6 - x}}{x^3 + 1} \approx \frac{3x^3}{x^3} \) as \( x \to \infty \).

Step 3: Evaluate the Limit

Using the simplified expression from Step 2, evaluate the limit: \( \lim_{x \to \infty} \frac{3x^3}{x^3 + 1} \). This simplifies to \( \frac{3x^3}{x^3} \) as the term \( +1 \) becomes negligible compared to \( x^3 \). Thus, the limit evaluates to \( \frac{3}{1} = 3 \).

Key Concepts

Dominant Terms in Limits

When calculating limits, especially as a variable approaches infinity, identifying the dominant terms becomes crucial. Dominant terms are those terms that have the greatest power or influence on the expression's behavior as the variable grows indefinitely large. This essentially boils down to picking the part of the expression that grows the fastest or has the largest impact.

In the given expression \(\frac{\sqrt{9x^6 - x}}{x^3 + 1}\), we focus first on the numerator \(\sqrt{9x^6 - x}\). As \(x\) becomes very large, the term \(9x^6\) will overwhelmingly dominate over the \(- x\). Therefore, we approximate \(\sqrt{9x^6} = 3x^3\) because \(9x^6\) is under a square root.

• Numerator Dominant Term: \(3x^3\)
• Denominator Dominant Term: \(x^3\)

Thus, finding which term primarily controls the behavior helps in simplifying our calculation greatly. Recognizing dominant terms allows us to streamline complex expressions and focus on only those that contribute most significantly to the value of the limit at infinity.

Infinity in Limits

The concept of infinity in limits is a fundamental aspect of calculus that deals with what happens to an expression as a variable increases indefinitely. Infinity isn't a number, but rather a description of boundlessness or an unending quantity. In our exercise, this concept helps us determine what happens to the function \(\frac{\sqrt{9x^6 - x}}{x^3 + 1}\) as \(x\) approaches infinity.

• Approach the problem by considering terms that dominate as \(x\) grows indefinitely, which helps in simplifying the limit evaluation process.
• Infinity in limits highlights which elements in the expression become negligible and which ones dictate the overall behavior as \(x\) becomes infinitely large.

As \(x\) moves towards infinity, terms like \(+1\) in the denominator become negligible compared to \(x^3\). Then, it is fairly straightforward to evaluate the simplified expression for its limit as it captures the significant contributors to the final outcome of the function.

Simplifying Expressions in Calculus

Simplifying expressions is an essential skill in resolving limits. In calculus, simplification involves reducing expressions to their most basic form without affecting their limits significantly, especially when working with infinity.

In evaluating the given limit, the cluster of terms can be reduced for simplification:

• Use the dominant terms identified: from \(\sqrt{9x^6 - x}\), utilize \(3x^3\) from the dominant \(\sqrt{9x^6}\).
• For the denominator, \(x^3 + 1\), \(x^3\) leads due to its dominance over the constant.

This leads to simplifying the function \(\frac{\sqrt{9x^6 - x}}{x^3 + 1} \approx \frac{3x^3}{x^3}\).
Such simplification helps in revealing the limit easily: \(\lim_{x \to \infty} \frac{3x^3}{x^3} = \frac{3}{1} = 3\), showcasing how simplification can effectively guide us to straightforward limit evaluation.