Problem 84 - Thomas Calculus

Question

Find the limits in Exercises \(80-86\). $$\lim _{x \rightarrow \infty}\left(\sqrt{9 x^{2}-x}-3 x\right)$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify Dominant Terms

When finding limits as \(x\) approaches infinity, it's crucial to identify the dominant terms within an expression. In this case, the dominant terms in \(9x^2 - x\) and \(3x\) are \(9x^2\) and \(3x\) respectively.

Step 2: Simplify Under Radical

Inside the square root, \(\sqrt{9x^2 - x}\), factor out \(x^2\), which is the dominant term:\[\sqrt{x^2\left(9 - \frac{1}{x}\right)} = x\sqrt{9 - \frac{1}{x}}.\] This simplification assumes \(x > 0\), which is valid as \(x\) approaches infinity.

Step 3: Express Limit as a Difference

Now express the limit as:\[ \lim_{x \to \infty} \left( x\sqrt{9 - \frac{1}{x}} - 3x \right). \] Factoring out \(x\) gives:\[ x \left( \sqrt{9 - \frac{1}{x}} - 3 \right). \]

Step 4: Evaluate the Limit Inside the Parenthesis

As \(x\) approaches infinity, \(\frac{1}{x}\) approaches zero, so:\[ \sqrt{9 - \frac{1}{x}} \to \sqrt{9} = 3. \] Thus, the expression inside the parenthesis \(\sqrt{9 - \frac{1}{x}} - 3\) approaches 0.

Step 5: Final Evaluation

The limit is now:\[ \lim_{x \to \infty} x(0) = 0. \] Therefore, the original limit evaluates to 0.

Key Concepts

Dominant Terms

When working with calculus limits, especially as a variable approaches infinity, it's essential to focus on the dominant terms of a function. Dominant terms play a crucial role because they determine the behavior of functions as they grow large. Consider an equation such as the one in this problem: \( \sqrt{9x^2 - x} - 3x \).

• In \( 9x^2 - x \), the dominant term is \( 9x^2 \), because it grows much faster than the \( -x \) term as \( x \) becomes very large.
• Similarly, in \( 3x \), the only term is \( 3x \) itself, which we consider as the dominant expression.

By identifying these key terms, we simplify complex expressions and focus on the parts that change significantly with larger values of \( x \).

Simplification

Simplifying a function or expression is a valuable technique in calculus. It helps make complex problems easier to manage, especially when determining limits. In the given exercise, the primary form that needs simplification is \( \sqrt{9x^2 - x} \).To simplify, extract \( x^2 \) out of the square root, since it's the dominant term. This simplifies the expression to:\[ \sqrt{x^2(9 - \frac{1}{x})} = x\sqrt{9 - \frac{1}{x}}. \]The reason this simplification is valid is because it assumes \( x \) is a positive number, which is true as \( x \) approaches infinity. This step simplifies the further calculations of limits, by narrowing down what really impacts the behavior of the function.

Infinite Limits

Infinite limits refer to the behavior of functions as they approach very large (or small) values. When we say \( x \rightarrow \infty \), we examine how functions behave as \( x \) becomes extremely large. In our specific exercise, we want to find:\[ \lim_{x \to \infty} \left( x\sqrt{9 - \frac{1}{x}} - 3x \right). \]A helpful insight in such problems is realizing how secondary terms like \( \frac{1}{x} \) vanish as \( x \) increases, reducing the expression inside the radical to \( \sqrt{9} = 3 \). This further transforms our expression:\[ x\left(\sqrt{9 - \frac{1}{x}} - 3\right) \to x(0) = 0. \] Hence, despite the initial complexity, identifying dominant terms and simplifying expressions makes it considerably easier to evaluate infinite limits. This understanding is crucial in calculus for evaluating the behavior of functions as they scale beyond typical numeric boundaries. By approaching infinity, we determine the resultant tendency of an expression or function.