Question

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

Short answer

The limit is \(-\infty\).

Step-by-step solution

Identify the Highest Power

To find the limit of a rational function as \( x \) approaches \(-\infty\), first identify the highest power of \( x \) in both the numerator and the denominator. In this expression, the highest power in the numerator \( x^3 + 2x^2 + 1 \) is \( x^3 \), and in the denominator \( x^2 - 5 \) is \( x^2 \).

Factor Out the Highest Power

Factor out the highest power of \( x \) found in the denominator from both the numerator and the denominator. This will help simplify the expression: \[ \lim_{x \rightarrow -\infty} \frac{x^2 (x + \frac{2}{x} + \frac{1}{x^2})}{x^2 (1 - \frac{5}{x^2})} \]

Simplify the Expression

After factoring, cancel the common \( x^2 \) terms in the numerator and the denominator: \[ \lim_{x \rightarrow -\infty} (x + \frac{2}{x} + \frac{1}{x^2}) / (1 - \frac{5}{x^2}) \]

Evaluate Individual Limits

Examine the components of the expression as \( x \to -\infty \). The terms \( \frac{2}{x} \) and \( \frac{1}{x^2} \) in the numerator and \( \frac{5}{x^2} \) in the denominator all approach zero. Thus, the expression simplifies to \( \lim_{x \rightarrow -\infty} x \), as the dominant term.

Determine the Limit

Since the remaining term in the expression is \( x \), and as \( x \) approaches \(-\infty\), this expression approaches \(-\infty\). Therefore, the limit is \(-\infty\).

Key concepts

Rational Functions

Rational functions are expressions that consist of two polynomials divided by each other. These functions often appear in mathematical problems involving limits, as in the given exercise. A rational function can generally be expressed in the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.

• The numerator \( P(x) \) is the polynomial "on top" of the fraction.
• The denominator \( Q(x) \) is the polynomial "under" the fraction.

In the given problem, the rational function is:\( \frac{x^3 + 2x^2 + 1}{x^2 - 5} \).

When approaching the concept of limits at infinity with rational functions, it's essential to focus on the highest power of \( x \) in both polynomials. These are the dominant terms that will dictate the behavior of the limit because they grow faster than the lower-powered terms as \( x \) approaches infinity or negative infinity.

Understanding how rational functions behave at their limits, particularly as \( x \) tends toward infinity, provides deep insights into the nature of their graphs. These insights include horizontal and asymptotic behaviors crucial for graph analysis.

Factoring Polynomials

Factoring polynomials is a key technique used to simplify expressions, particularly rational functions. By extracting the highest power of \( x \) from both the numerator and the denominator, the expression becomes easier to handle.

For the given rational function:\( \frac{x^3 + 2x^2 + 1}{x^2 - 5} \),
We need to factor out \( x^2 \) — the highest power of \( x \) present in the denominator, from both the numerator and denominator.

This results in:

• The numerator becoming: \( x^2(x + \frac{2}{x} + \frac{1}{x^2}) \).
• The denominator becoming: \( x^2(1 - \frac{5}{x^2}) \).

Once factored, the common factor \( x^2 \) is canceled out between the numerator and denominator, simplifying the expression:\( (x + \frac{2}{x} + \frac{1}{x^2}) / (1 - \frac{5}{x^2}) \).

This simplification makes it straightforward to analyze the expression as \( x \to -\infty \). Knowing how to factor and simplify polynomials is helpful not only in solving limits but also in various other calculus problems.

Analyzing Dominant Terms

In the context of limits at infinity, dominant terms are the terms in the polynomials that determine the overall behavior of the function. These terms are powered higher than any others in their respective polynomials. As a result, they increase or decrease the fastest as \( x \) tends towards infinity or negative infinity.

For the expression\( \frac{x + \frac{2}{x} + \frac{1}{x^2}}{1 - \frac{5}{x^2}} \),
The dominant term in the numerator is \( x \),
and in the denominator, it's the constant term \( 1 \), as other terms like \( \frac{2}{x} \) and \( \frac{1}{x^2} \) become negligible as \( x \to -\infty \).

By focusing on these dominant terms, others being negligible, the limit reduces to:\( \lim_{x \rightarrow -\infty} x \),
which approaches \(-\infty\).

Recognizing and analyzing dominant terms aid in efficiently solving limits, providing an understanding of the behavior of rational functions as \( x \) grows exceedingly large (either positively or negatively). Excluding insignificant terms simplifies calculations, making the path to finding the right answer clearer.