Problem 18 - Calculus of a Single Variable

Question

Finding Limits at Infinity In Exercises \(15-18\) , find each limit, if possible. $$ \begin{array}{l}{\text { (a) } \lim _{x \rightarrow \infty} \frac{5 x^{3 / 2}}{4 x^{2}+1}} \\ {\text { (b) } \lim _{x \rightarrow \infty} \frac{5 x^{3 / 2}}{4 x^{3 / 2}+1}} \\ {\text { (c) } \lim _{x \rightarrow \infty} \frac{5 x^{3 / 2}}{4 \sqrt{x}+1}}\end{array} $$

Step-by-Step Solution

Answer

The limits for (a), (b) and (c) are \(0\), \(\frac{5}{4}\) and \(\infty\) respectively.

Step 1: Identify the degree of polynomials and apply the rule

Recognize that the fraction parts in (a), (b), and (c) are rational functions, i.e., ratios of polynomials. Apply the rule for limits at infinity of rational functions: If the denominator's degree is larger than the numerator's, the limit is 0. If both have the same degree, the limit is the ratio of the leading coefficients. If the numerator's degree is larger, the limit is \(\pm \infty\) depending on the sign.

Step 2: Solve the limit for (a)

Here, both the numerator and the denominator are polynomials, with the numerator of degree \(3/2\) and the denominator of degree 2. So, the denominator's degree is larger, meaning the limit is 0.

Step 3: Solve the limit for (b)

In this case, both the numerator and denominator have the same degree, \(3/2\). Hence, the limit is the ratio of the leading coefficients, which is \(\frac{5}{4}\).

Step 4: Solve the limit for (c)

Here, the numerator's degree is \(3/2\), while the denominator's degree is \(1/2\). This means that the numerator's degree is larger resulting to the limit being \(\infty\).

Key Concepts

Polynomials

Polynomials are expressions that consist of variables raised to various powers, summed with coefficients. They are fundamental objects in algebra and calculus.
For example, consider the polynomial expression \(2x^3 + 3x^2 - x + 5\). In this expression:

• Each term is made up of a power of \(x\) and a corresponding coefficient, like \(2x^3\).
• The highest power of \(x\) in this expression is the degree of the polynomial.

Understanding polynomials is essential for solving problems involving limits at infinity.
When calculating limits, especially at infinity, recognizing the most significant term (highest power of \(x\)) gives us vital insights.

Rational Functions

A rational function is a ratio of two polynomials. It looks like a fraction where both the numerator and the denominator are polynomials. For instance:
\[R(x) = \frac{p(x)}{q(x)}\] Here, \(p(x)\) and \(q(x)\) are polynomials. Rational functions are quite common in calculus as they allow us to understand various behaviors of graphs and limits.

• If the degree of \(p(x)\) is larger than the degree of \(q(x)\), the function tends toward infinity as \(x\) grows larger.
• If both have the same degree, the limit depends on the ratios of their leading coefficients.
• If \(q(x)\) has a higher degree, the function tends towards zero.

These rules help us to predict and compute limits at infinity effectively, as used in finding solutions for limits like \(\lim_{x \to \infty}\frac{5x^{3/2}}{4x^2+1}\).

Degree of Polynomials

The degree of a polynomial is the highest power of the variable \(x\) in the polynomial. Knowing the degree of a polynomial is crucial for evaluating limits, especially at infinity.
For example, in a polynomial \(3x^2 + 2x + 5\), the degree is 2, as \(x\) is raised to the power of two in the term \(3x^2\).
When comparing degrees in rational functions:

• If the degree of the numerator is higher, the function trends towards infinity.
• If the degree of the denominator is higher, the function trends towards zero.
• If both degrees are equal, the limit results in a constant, derived from the leading coefficients.

These degree comparisons allow us to apply rules for limits at infinity, solving equations efficiently.

Leading Coefficients

Leading coefficients in polynomials are the numbers that multiply the term with the highest power in the expression.
For instance, in the polynomial \(7x^4 - 2x^2 + 3\), the leading coefficient is 7, since \(x^4\) is the term with the highest degree.
When working with rational functions and their limits at infinity:

• If the degrees of the numerator and denominator match, the leading coefficient determines the limit.
• The limit becomes the ratio of these leading coefficients.
• This ratio represents the trend of the function as \(x\) approaches infinity.

For instance, in the problem \(\lim_{x \to \infty}\frac{5x^{3/2}}{4x^{3/2}+1}\), both polynomial degrees are \(3/2\), making the limit \(\frac{5}{4}\). Understanding leading coefficients simplifies solving and predicting these types of limit problems.