Problem 10 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Dominant Terms

When considering the limit as \( x \to -\infty \), we need to identify the dominant terms in both the numerator and the denominator. For the numerator \( x^2 + 2 \), the dominant term is \( x^2 \). For the denominator \( x^3 + x + 1 \), the dominant term is \( x^3 \).

Step 2: Rewrite the Expression with Dominant Terms

Rewrite the original expression \( \frac{x^2 + 2}{x^3 + x + 1} \), focusing on the dominant terms identified earlier. This gives us the simplified expression \( \frac{x^2}{x^3} \).

Step 3: Simplify the Expression

Simplify \( \frac{x^2}{x^3} \) by canceling out common powers of \( x \). The expression simplifies to \( \frac{1}{x} \).

Step 4: Evaluate the Simplified Limit

Determine the limit of the simplified expression \( \frac{1}{x} \) as \( x \to -\infty \). As \( x \) becomes very large negatively, \( \frac{1}{x} \) approaches 0.

Key Concepts

Dominant Terms

When evaluating limits, especially as the variable approaches very large numbers or infinity, identifying the dominant terms becomes crucial. In an expression like \( \frac{x^2 + 2}{x^3 + x + 1} \), dominant terms are the ones that grow fastest as \( x \) approaches infinity or negative infinity.

• For the numerator \( x^2 + 2 \), the dominant term is \( x^2 \) because \( x^2 \) will grow much larger than the constant 2 as \( x \) increases in magnitude.
• In the denominator \( x^3 + x + 1 \), the dominant term is \( x^3 \) since it grows faster than both \( x \) and the constant 1.

Identifying these dominant terms allows us to focus on the most significant components of the expression when \( x \) is very large or very small, simplifying our limit evaluation.

Simplifying Expressions

Simplifying expressions, especially when dealing with limits, is all about making complex expressions easier to work with by keeping only the essential parts. Once we've identified the dominant terms, the next step is restructuring the expression while keeping the dominant terms at the forefront.

To simplify \( \frac{x^2 + 2}{x^3 + x + 1} \):

• Focus on \( \frac{x^2}{x^3} \), as these are the dominant terms both in the numerator and the denominator.
• The expression simplifies to \( \frac{1}{x} \) after canceling the common term \( x^2 \) from both the top and bottom, transforming it into a much simpler form.

This simplification step is vital, as it translates a complex limit problem into a straightforward one by focusing our attention on the terms that matter the most.

Evaluating Limits

Once the expression is simplified, evaluating the limit becomes straightforward. The simplified form of the expression \( \frac{1}{x} \) is now ready for limit evaluation as \( x \to -\infty \).

Here’s how you evaluate:

• As \( x \to -\infty \), the denominator \( x \) becomes very large negatively, meaning \( \frac{1}{x} \) approaches zero.
• The negative sign indicates the direction of infinity, but since \( \frac{1}{x} \) only depends on approaching large magnitudes, the ultimate result remains zero.

By understanding how \( 1/x \) behaves as \( x \to -\infty \), we conclude that the limit is 0. Evaluating limits in this way uses logic and mathematics to predict behavior as \( x \) moves towards extreme values, providing a deeper understanding of how functions behave.