Question

Find \(\lim _{x \rightarrow \infty} h(x)\), if possible. \(f(x)=5 x^{3}-3\) (a) \(h(x)=\frac{f(x)}{x^{2}}\) (b) \(h(x)=\frac{f(x)}{x^{3}}\) (c) \(h(x)=\frac{f(x)}{x^{4}}\)

Short answer

The limits of \(h(x)\) as \(x\) approaches infinity for each part are: (a) \(5x\), (b) \(5\), (c) \(0\).

Step-by-step solution

Simplify \(h(x)\)

Given the function \(f(x)=5x^{3}-3\), when we substitute \(f(x)\) into \(h(x)\) we get: (a) \(h(x)=\frac{5x^{3}-3}{x^{2}}\), (b) \(h(x)=\frac{5x^{3}-3}{x^{3}}\), (c) \(h(x)=\frac{5x^{3}-3}{x^{4}}\).

Simplify the Fractions

Let's simplify each part \(h(x)\). (a) \(h(x)=5x- \frac{3}{x^{2}}\), (b) \(h(x)=5-\frac{3}{x^{3}}\), (c) \(h(x)=5x^{-1}-3x^{-4}\).

Apply the Limit Rules

Apply the limit rules to find \(\lim _{x \rightarrow \infty} h(x)\) for each part. As \(x\) approaches infinity, any term with \(x\) in the denominator approaches 0. Therefore, (a) \(\lim _{x \rightarrow \infty} h(x)=5x- \frac{3}{x^{2}} = 5x-0 = 5x\), (b) \(\lim _{x \rightarrow \infty} h(x)=5-\frac{3}{x^{3}} = 5-0 = 5\), (c) \(\lim _{x \rightarrow \infty} h(x)=5x^{-1}-3x^{-4} = 0-0 = 0\).

Key concepts

Polynomial Functions

Polynomial functions are a fundamental part of calculus and algebra. They are expressions that consist of variables raised to whole number exponents, each multiplied by a coefficient. A typical polynomial function can be written as \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where \( a_n eq 0 \) unless the polynomial is a constant.

• The degree of the polynomial is determined by the highest exponent (the power of \( x \)) in the expression. For instance, the function \( f(x) = 5x^3 - 3 \) is a cubic polynomial, which means it has a degree of 3, because the highest power of \( x \) is 3.
• Polynomial functions are continuous and differentiable, meaning they can produce a graph that is a smooth curve without any breaks or jumps.
• They have predictable shapes depending on their degree: linear polynomials create lines, quadratics form parabolas, cubic polynomials produce a classic "S" shape, and so on.

To find the behavior of a polynomial as \( x \rightarrow \infty \), one usually looks at the leading term, as it will dominate the behavior of the polynomial for large values of \( x \).

Rational Functions

Rational functions are formed by dividing one polynomial by another. In general, a rational function can be expressed as \( h(x) = \frac{f(x)}{g(x)} \), where both \( f(x) \) and \( g(x) \) are polynomials, and \( g(x) eq 0 \).

• Rational functions can exhibit a variety of behaviors not seen in simple polynomials, such as vertical asymptotes (where the function approaches infinity) and horizontal asymptotes (which indicate a leveling out as \( x \) approaches infinity), depending on the degrees of the numerator and denominator.
• When dealing with rational functions and limits, simplifying the function by factoring and reducing terms is often key to identifying asymptotic behavior, particularly as \( x \rightarrow \infty \).
• For example, in the provided problem, \( h(x) = \frac{5x^3 - 3}{x^n} \) varies in behavior depending on the value of \( n \). Reducing the function to its simplest form helps to understand how individual terms impact the overall function as \( x \) increases.

Behavior at Infinity

Understanding a function's behavior at infinity is crucial in calculus, as it helps us comprehend how functions behave in extreme cases, which aids in predicting and analyzing real-world phenomena. This concept is especially important for limits.

• When we say \( \lim_{x \to \infty} h(x) \), we are trying to find out what value, if any, \( h(x) \) approaches as \( x \) becomes very large.
• When simplifying rational functions, terms with \( x \) in the denominator tend to contribute less as \( x \) becomes very large. For instance, terms like \( \frac{3}{x^2} \) tend to zero, heavily influencing the computation of the limit.
• This behavior means that for the problem at hand:
  • In (a), as \( x \rightarrow \infty \), \( 5x - \frac{3}{x^2} \rightarrow 5x \), showing that the linear term dominates.
  • In (b), \( 5 - \frac{3}{x^3} \rightarrow 5 \), since \( 5 \) is constant and the fraction diminishes.
  • In (c), \( 5x^{-1} - 3x^{-4} \rightarrow 0 \), indicating both terms reduce to zero.

The larger the power in the denominator, the faster a term will shrink towards zero, thus controlling which terms influence the limit as \( x \) approaches infinity.