Question

Evaluate the given limit. $$ \lim _{x \rightarrow \infty} \frac{x^{3}+2 x^{2}+1}{5-x} $$

Short answer

The limit is infinity \((\infty)\).

Step-by-step solution

Analyze the Dominant Terms

The dominant term in the numerator is the highest-degree term, which is \(x^3\). In the denominator, the highest-degree term is the linear term \(x\). These dominant terms will predominantly influence the behavior as \(x\) approaches infinity.

Simplify the Expression

We simplify the fraction by dividing every term in the numerator and the denominator by \(x^3\), which is the highest power in the expression:\[\frac{x^3 + 2x^2 + 1}{5 - x} = \frac{x^3(1 + \frac{2}{x} + \frac{1}{x^3})}{x(-\frac{5}{x} + 1)}\]

Evaluate the Limit by Substituting Dominant Behavior

Substitute the dominant behavior as \(x\) goes to infinity. The numerator simplifies to \(1 + \frac{2}{x} + \frac{1}{x^3} \approx 1\) since terms with \(\frac{1}{x}\) will approach zero. In the denominator, we approximate \(-\frac{5}{x} + 1 \approx 1\):\[\lim_{x \to \infty} \frac{x^3 (1 + \frac{2}{x} + \frac{1}{x^3})}{x(1)}\] which simplifies to:\[\lim_{x \to \infty} \frac{x^3}{x} = \lim_{x \to \infty} x^2\]

Evaluate the Limit of \(x^2\)

Since \(x^2\) grows without bound as \(x\) goes to infinity, the limit is infinity:\[\lim_{x \to \infty} x^2 = \infty\]

Key concepts

Limits

Limits are a fundamental concept in calculus and help us understand how functions behave as they approach a certain point or infinity. In this exercise, we are dealing with the limit as \(x\) approaches infinity, specifically looking at how the function \(\frac{x^3 + 2x^2 + 1}{5 - x}\) behaves. To evaluate this limit, we analyze how the function behaves as the variable \(x\) becomes very large. It allows us to see the trend or pattern rather than computing the exact function value at infinity, which is not possible. Understanding limits helps in various fields like engineering, physics, and economics to predict long-term behavior of different models. Thus, limits provide a powerful tool for identifying trends in mathematical functions and help bridge calculus with real-world applications.

Dominant Term Analysis

When evaluating limits, especially at infinity, the terms with the highest degree in the numerator and the denominator significantly influence the outcome. This is known as dominant term analysis. In the given problem, the highest power of \(x\) in the numerator is \(x^3\), while in the denominator, it's \(x\). These terms become the dominant terms. Why focus on dominant terms?

• The other terms become insignificant as \(x\) grows larger.
• They primarily dictate the behavior of the entire expression as \(x\) approaches infinity.

By concentrating on these, we simplify our calculations, making the problem more manageable and helping us quickly identify the trend or behavior of the function.

Simplification of Expressions

Simplifying expressions makes complex mathematical problems easier to solve. When dealing with limits, simplifying the expression helps reveal the essential characteristics of the function. In this case, the expression \(\frac{x^3 + 2x^2 + 1}{5 - x}\) is simplified by dividing each term by the highest degree of \(x\), which is \(x^3\). This reduces the complex expression into a simpler form, emphasizing the dominant behavior:\[\frac{x^3(1 + \frac{2}{x} + \frac{1}{x^3})}{x(1)}\]This step is crucial because it:

• Makes it clear which terms grow or shrink as \(x\) approaches infinity.
• Facilitates the identification of terms that tend towards zero, allowing for clear evaluation of the limit.

Simplification is a key strategy in calculus, reducing the complexity of expressions to make analyses manageable.

Infinite Limits

Infinite limits occur when we evaluate the behavior of a function as it approaches infinity. This particular limit evaluates \(\lim_{x \to \infty} x^2\), concluding that it grows indefinitely as \(x\) becomes infinitely large. Key insights into infinite limits include:

• Not all functions approach a specific number as \(x\) goes to infinity.
• Some functions increase without bound, like \(x^2\) in our exercise.

Understanding infinite limits helps in fields requiring knowledge of asymptotic behavior and trends over extended period or ranges, such as theoretical physics and computational algorithms. Appreciating these concepts gives students the analytical tools to study functions that don't settle at finite values but exhibit persistent growth or decrease.