Problem 7 - Precalculus: Mathematical for Calculus

Question

Find the limit. $$\lim _{x \rightarrow-\infty} \frac{4 x^{2}+1}{2+3 x^{2}}$$

Step-by-Step Solution

Answer

The limit is \(\frac{4}{3}\).

Step 1: Identify Dominant Terms

For limits involving rational expressions as \(x\) approaches \(-\infty\), identify the dominating terms in the numerator and the denominator. In this case, the dominating terms in \(4x^2 + 1\) and \(2 + 3x^2\) are \(4x^2\) and \(3x^2\), respectively, because they have the highest powers of \(x\).

Step 2: Simplify the Expression

Divide every term in both the numerator and the denominator by \(x^2\), the highest power of \(x\) present in the expression: \[ \lim_{x \to -\infty} \frac{4x^2 + 1}{2 + 3x^2} = \lim_{x \to -\infty} \frac{x^2(4 + \frac{1}{x^2})}{x^2(\frac{2}{x^2} + 3)} \] This simplifies to: \[ \lim_{x \to -\infty} \frac{4 + \frac{1}{x^2}}{\frac{2}{x^2} + 3} \]

Step 3: Evaluate the Limit

As \(x\) approaches \(-\infty\), the terms \(\frac{1}{x^2}\) and \(\frac{2}{x^2}\) approach 0 because \(\frac{1}{x^2}\) becomes very small. Thus: \[ \lim_{x \to -\infty} \frac{4 + \frac{1}{x^2}}{\frac{2}{x^2} + 3} = \lim_{x \to -\infty} \frac{4 + 0}{0 + 3} = \frac{4}{3} \]

Key Concepts

Rational expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They involve division of one polynomial expression by another. These expressions often look complex, but they follow the same basic rules as ordinary fractions.
In mathematical terms, a rational expression is an expression of the form \( \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials. Understanding how to simplify these fractions is crucial for dealing with them effectively, especially as we analyze their behavior at infinity.

• When simplifying rational expressions, try to factorize the numerator and the denominator to see if you can cancel out any common factors.
• Be aware that division by zero is undefined, so identify any values of the variable that would make the denominator zero, as these indicate points of discontinuity.

In the exercise provided, we recognize that the rational expression is \( \frac{4x^2 + 1}{2 + 3x^2} \), which involves polynomials in both the numerator and the denominator.

Dominant terms in polynomials

Identifying the dominant terms in polynomials is a key step in determining limits at infinity. The dominant term is the term with the highest power of \( x \), since it tends to grow faster than the other terms as \( x \) becomes very large or very small.
This concept allows us to simplify complex expressions by focusing on these key terms and treating other terms as negligibly small when compared to the dominant term.
In our exercise, the polynomial \( 4x^2 + 1 \) has a dominant term of \( 4x^2 \), while \( 2 + 3x^2 \) has a dominant term of \( 3x^2 \).

• When dealing with limits involving rational expressions, always look for the term with the highest degree of \( x \) in both the numerator and the denominator.
• This dominant term helps us simplify the expression by dividing all terms by \( x^2 \), as shown in the exercise.

This process effectively reduces the rational expression to its simplest form that reveals how the expression behaves as \( x \) approaches negative infinity.

Limits at infinity

Limits at infinity involve analyzing what happens to a function as the input, \( x \), becomes very large (positive or negative). This is particularly useful in understanding the end behavior of functions and rational expressions.
In our exercise, we are finding the limit as \( x \) approaches \(-\infty\). This means we're interested in the behavior of \( \frac{4x^2 + 1}{2 + 3x^2} \) as \( x \) becomes a very large negative number.

• To evaluate limits at infinity for rational expressions, divide each term by the highest power of \( x \) in the denominator. This simplifies the expression significantly.
• Analyze what happens to the terms as \( x \) becomes very large. Typically, rational terms like \( \frac{1}{x^n} \) (where \( n > 0 \)) approach 0.

In the provided solution, applying these steps allowed for the simplification \( \lim_{x \to -\infty} \frac{4 + \frac{1}{x^2}}{\frac{2}{x^2} + 3} = \frac{4}{3} \), as negligible terms vanish due to approaching zero.