Problem 31 - Single Variable Calculus

Question

Find the limit. $$ \lim _{x \rightarrow \infty} \frac{2 x}{\sqrt{3 x^{2}+1}} $$

Step-by-Step Solution

Answer

The limit of the given function as x approaches infinity is 0: \[ \lim_{x \rightarrow \infty} \frac{2x}{\sqrt{3x^2 + 1}} = 0 \]

Step 1: Identify the largest terms in both numerator and denominator

In the numerator, the largest term is \(2x\). In the denominator, the largest term is \(\sqrt{3x^2}\) as x approaches infinity.

Step 2: Divide every term in the function by the largest term in the denominator

Divide the numerator and denominator by \(\sqrt{3x^2}\), the whole equation would then look like: \[ \lim_{x \rightarrow \infty} \frac{\frac{2x}{\sqrt{3x^2}}}{\frac{\sqrt{3x^2 + 1}}{\sqrt{3x^2}}} \] We can simplify the expression by dividing both terms in the numerator: \[ \lim_{x \rightarrow \infty} \frac{\frac{2x}{\sqrt{3x^2}}}{\frac{\sqrt{3x^2 + 1}}{\sqrt{3x^2}}} = \lim_{x \rightarrow \infty} \frac{2x}{3x^2} \cdot \frac{\sqrt{3x^2}}{\sqrt{3x^2 + 1}} \]

Step 3: Simplify the limit expression

Now, let's simplify the limit expression: \[ \lim_{x \rightarrow \infty} \frac{2x}{3x^2} \cdot \frac{\sqrt{3x^2}}{\sqrt{3x^2 + 1}} = \lim_{x \rightarrow \infty} \frac{2}{3x} \cdot \frac{\sqrt{3x^2}}{\sqrt{3x^2 + 1}} \]

Step 4: Evaluate the limit expression

As x approaches infinity, the term \(\frac{2}{3x}\) approaches 0. Therefore, the limit expression becomes: \[ \lim_{x \rightarrow \infty} \frac{2}{3x} \cdot \frac{\sqrt{3x^2}}{\sqrt{3x^2 + 1}} = 0 \cdot \lim_{x \rightarrow \infty} \frac{\sqrt{3x^2}}{\sqrt{3x^2 + 1}} \]

Step 5: Final Answer

Since multiplying by 0 makes the entire expression equal to 0, the limit is: \[ \lim_{x \rightarrow \infty} \frac{2x}{\sqrt{3x^2 + 1}} = 0 \] The limit of the given function as x approaches infinity is 0.

Key Concepts

Understanding Infinity Limits

When working with limits in calculus, especially as a variable approaches infinity, we analyze how the function behaves at values that grow larger and larger. These infinity limits help us understand long-term behavior of functions and are crucial for dealing with functions that increase without bound.

The concept of infinity limits can be quite abstract. Infinity is not a number but a direction describing endless growth in the positive or negative sense. Because of this, when expressing limits as variables approach infinity, we focus on the dominant terms that have the most significant impact at large scale.

In the example above, as \( x \) approaches infinity, the task was to find the behavior of \( \frac{2x}{\sqrt{3x^2 + 1}} \). The numerator and the denominator were analyzed separately to determine which terms were the most influential. The largest terms guide our understanding because smaller terms contribute progressively less as \( x \) grows large. Therefore, we don't just let \( x \) explode to infinity; instead, we observe how the main components of the function relate to each other in this process.

Limit Simplification Techniques

Simplifying limits is an essential step in finding their values, especially when dealing with rational functions. Rational functions are ratios of polynomials, and they often require simplification to evaluate limits that involve infinity.

In the provided solution, simplification involved dividing terms by the largest term in the denominator, which is a common technique. This approach focuses on reducing the expression to component forms that clarify the long-term behavior of the function. Each term was divided by \( \sqrt{3x^2} \) because it simplifies the squareroot term into an easier-to-manage form as \( x \) heads to infinity.

• Identify the terms with the highest power in both the numerator and the denominator.
• Divide each term by the largest term's counterpart to simplify the problem.
• Reevaluate the expression to focus only on terms significant at infinity.

By simplifying the terms, we were able to approach a cleaner and more intuitive form of the original problematic expression, showcasing that each individual component tends towards a simple behavior. As shown, \( \frac{2x}{\sqrt{3x^2}} \) becomes \( \frac{2}{3x} \), indicating that as \( x \) approaches infinity, the expression itself moves towards 0.

Rational Functions and Their Limits

Rational functions are composed of polynomials in their numerator and denominator, and analyzing their limits as variables approach infinity is crucial in calculus. When these functions are involved, simplification becomes a key technique to determine the behavior of the function at infinity.

Often, these functions demonstrate horizontal asymptotic behaviors at infinity, meaning they approach a horizontal line as variables become infinitely large. Understanding the degree of polynomials in the function's components is necessary to predict its long-term behavior.

• If the degree of the polynomial in the numerator is less than that in the denominator, the function often trends towards zero.
• If the numerator's degree is similar to the denominator's, it typically approaches a constant ratio.
• If the numerator's degree exceeds the denominator's degree, the function typically trends towards infinity or negative infinity.

In the exercise given, \( \frac{2x}{\sqrt{3x^2 + 1}} \), it's observed that the polynomial degree of \( x \) in the simplified form reveals that it trends towards zero at infinity due to the larger degree in the denominator, demonstrating the expected rational function behavior.