Problem 12 - Calculus

Question

In Exercises \(10-15,\) give \(\lim _{x \rightarrow-\infty} f(x)\) and \(\lim _{x \rightarrow+\infty} f(x)\). $$f(x)=x^{5}+25 x^{4}-37 x^{3}-200 x^{2}+48 x+10$$

Step-by-Step Solution

Answer

\( \lim_{x \to -\infty} f(x) = -\infty \), \( \lim_{x \to +\infty} f(x) = +\infty \).

Step 1: Identify the Leading Term

The function given is \( f(x) = x^5 + 25x^4 - 37x^3 - 200x^2 + 48x + 10 \). The leading term is the term with the highest power of \( x \), which is \( x^5 \). This term will dominate the behavior of the function as \( x \to \pm \infty \).

Step 2: Evaluate \( \lim_{x \to -\infty} f(x) \)

Consider the leading term \( x^5 \). As \( x \to -\infty \), since 5 is an odd exponent, \( x^5 \to -\infty \) as well. This suggests \( \lim_{x \to -\infty} f(x) = -\infty \).

Step 3: Evaluate \( \lim_{x \to +\infty} f(x) \)

Using the leading term \( x^5 \) again, as \( x \to +\infty \), \( x^5 \to +\infty \). Thus, the behavior of the function is dictated by this leading term, indicating \( \lim_{x \to +\infty} f(x) = +\infty \).

Key Concepts

Polynomial Function

A polynomial function is an essential concept in mathematics, particularly in algebra and calculus. At its core, a polynomial function is any function that can be expressed in the form:

• \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \)

Here, the coefficients \(a_n, a_{n-1}, \ldots, a_0\) are real numbers, and the powers of \(x\) are whole numbers, meaning they can't be negative or fractions. The highest power of \(x\) present is known as the degree of the polynomial, such as \(n\) in this expression, which greatly influences the function's shape and behavior.
Common examples include linear functions (degree 1), quadratic functions (degree 2), and our example in this context is a degree 5 polynomial function. These functions can model a variety of real-world behaviors, which is why they are so highly valued in mathematical analysis.

Leading Term

The leading term of a polynomial function is crucial because it determines the overall trend or direction of the graph as \(x\) approaches infinity or negative infinity. It's defined as the term in the polynomial that has the highest degree (or the highest exponent on the variable \(x\)).

• In the function \(f(x) = x^5 + 25x^4 - 37x^3 - 200x^2 + 48x + 10\), the leading term is \(x^5\).

This term is essential because it "leads" the way the function behaves towards the double infinity, despite the other terms' values. As \(x\) becomes very large positively or negatively, the other terms grow insignificant compared to the leading term. This understanding simplifies calculations and predictions concerning the function's growth as it clearly shows what the function behaves like at these extremes.
Hence, when analyzing limits at infinity, focusing on the leading term provides insights into how the function will behave when \(x\) is considerably large in magnitude.

End Behavior

End behavior in the context of polynomial functions refers to what happens to the value of the function as the input \(x\) moves towards positive or negative infinity. Understanding this helps in predicting the function's long-term trends, which is incredibly useful for graphing and analysis. For any polynomial, the end behavior is dictated primarily by its leading term:

• If the degree is odd and the leading coefficient is positive, as \(x \to \infty\), \(f(x) \to \infty\) and as \(x \to -\infty\), \(f(x) \to -\infty\).
• If the degree is odd and the leading coefficient is negative, both ends switch directions.
• Conversely, with even degrees, both ends of the graph point either both upwards or both downwards depending on the leading coefficient's sign.

For our polynomial \(f(x) = x^5 + \ldots\), since the degree is 5 (odd) and the coefficient of \(x^5\) is positive, the end behavior is such that \(\lim_{x \to \infty} f(x) = \infty\) and \(\lim_{x \to -\infty} f(x) = -\infty\). This characteristic arm pattern (opposing directions) is seen commonly in odd-degree polynomials, further illustrating the value of understanding leading terms and their consequent impact on end behavior.