Problem 23 - Single Variable Calculus

Question

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

Step-by-Step Solution

Answer

The limit as x approaches negative infinity of the given function \(\frac{1-2 x^{2}}{x^{3}+1}\) is 0.

Step 1: Identify the degrees of the polynomials in the numerator and denominator.

The given function is: \[ \frac{1-2 x^{2}}{x^{3}+1} \] The degrees of the polynomials are: - Degree of the numerator = 2 - Degree of the denominator = 3

Step 2: Determine the behavior of the function as x approaches negative infinity based on the degrees.

As x approaches negative infinity, the term with the highest power in the numerator and denominator will dominate the behavior of the function. Since the degree of the denominator is higher than the degree of the numerator (3 > 2), the rational function will approach zero.

Step 3: Calculate the limit.

As we concluded in step 2, the limit of the given function as x approaches negative infinity is 0: \[ \lim _{x \rightarrow-\infty} \frac{1-2 x^{2}}{x^{3}+1} = 0 \] Hence, the limit as x approaches negative infinity is 0.

Key Concepts

Rational Functions

Rational functions are expressions that result from dividing two polynomials. They can come in handy for solving limits, as they offer insight into the behavior of functions. Understanding rational functions involves analyzing the numerator and the denominator separately. The function given here, \( \frac{1-2x^2}{x^3+1} \), is a clear example of a rational function since both the numerator and the denominator are polynomials.

• The numerator: \( 1-2x^2 \)
• The denominator: \( x^3+1 \)

Each polynomial has distinct degrees, which are the highest powers of \( x \) in each polynomial. The relative degrees of polynomials then play a crucial role in how the rational function behaves, especially when considering limits.

Degrees of Polynomials

The degree of a polynomial is simply the highest power of its variable, which dictates how fast it grows as the variable increases. In the exercise, we examine:

• Numerator \( 1-2x^2 \): Degree is 2 (due to \( x^2 \))
• Denominator \( x^3+1 \): Degree is 3 (due to \( x^3 \))

Knowing the degrees allows us to predict which terms dominate as \( x \) grows larger or smaller. When comparing the two polynomials, we observe where the degrees differ. Here, the denominator's degree (3) is greater than that of the numerator (2), which is vital for understanding the limit's behavior at infinity.

Behavior of Functions at Infinity

Determining the limit of a function as \( x \) approaches infinity (or negative infinity) involves recognizing how the leading terms of polynomials influence the entire expression. In the given rational function \( \frac{1-2x^2}{x^3+1} \), the higher degree in the denominator (3) compared to the numerator (2) indicates that the denominator grows faster than the numerator as \( x \) moves towards negative infinity.
Here's what happens:

• As \( x \rightarrow -\infty \), \( x^3 \) becomes very large negatively compared to \( 2x^2 \).
• This fast-growing denominator leads the whole fraction towards zero because the larger denominator "overpowers" the numerator.

Thus, the behavior of the rational function at negative infinity suggests that the function approaches a limit of 0, as calculated in the exercise. This behavior is typical when the degree of the numerator is lower than that of the denominator.