Problem 10 - Calculus

Question

In Exercises \(10-15,\) give \(\lim _{x \rightarrow-\infty} f(x)\) and \(\lim _{x \rightarrow+\infty} f(x)\). $$f(x)=-x^{4}$$

Step-by-Step Solution

Answer

Both limits are \(-\infty\): \( \lim_{x \rightarrow -\infty} f(x) = -\infty \) and \( \lim_{x \rightarrow +\infty} f(x) = -\infty \).

Step 1: Understand the function

The function given is \( f(x) = -x^4 \). This is a polynomial function with a highest degree term of \( -x^4 \).

Step 2: Analyze behavior as x approaches negative infinity

As \( x \) approaches negative infinity, \( x^4 \) becomes very large since raising a negative number to an even power yields a positive result. However, since there is a negative sign in front of \( x^4 \), \( -x^4 \) will approach negative infinity \((-\infty)\).

Step 3: Analyze behavior as x approaches positive infinity

As \( x \) approaches positive infinity, \( x^4 \) also becomes very large. Since the function \( f(x) = -x^4 \) includes a negative sign, \( -x^4 \) will again approach negative infinity \((-\infty)\).

Step 4: Conclude the limits

Therefore, \( \lim_{x \rightarrow -\infty} f(x) = -\infty \) and \( \lim_{x \rightarrow +\infty} f(x) = -\infty \).

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression that consists of variables raised to different power levels, usually combined using addition, subtraction, and multiplication. The basic structure of a polynomial is given by the general formula:\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \]where \( a_n \) to \( a_0 \) are constants, and \( n \) indicates the degree of the polynomial. When thinking about polynomial functions, consider these key points:

• The degree of the polynomial, which is the highest power of \( x \) that appears in it, determines the most significant behavior of the function.
• Polynomial functions are continuous and smooth curves, meaning they have no breaks or sharp points.
• The leading term (the term with the highest degree) significantly influences the behavior and shape of the graph of the function.

Understanding polynomial functions helps reveal a lot about their limits and behavior as \( x \) approaches infinity or negative infinity.

Limits at Infinity

Limits at infinity help us understand the behavior of functions as the variable \( x \) grows larger or smaller indefinitely. In this context, large means approaching infinity, and small means approaching negative infinity. When we talk about limits at infinity, we're interested in what the function values are doing as \( x \) moves increasingly far in either direction on the number line.

• For polynomial functions like \( f(x) = -x^4 \), as \( x \) approaches positive infinity (\( +\infty \)), the term \( x^4 \) becomes very large, leading the function towards its limit at \( -\infty \), due to the negative sign.
• Similarly, as \( x \) approaches negative infinity (\( -\infty \)), \( x^4 \) still becomes very large (since any negative number raised to an even power is positive), and thus \( f(x) \) tends again towards \( -\infty \).

This analysis of limits at infinity provides crucial insights, particularly when considering functions with multiple terms. Often, it is the highest degree term, here \( -x^4 \), which dictates the behavior as \( x \) moves towards infinite limits.

End Behavior

The term 'end behavior' describes how a function behaves as \( x \) approaches the extreme ends of the number line; that is, as \( x \) goes to positive infinity (\( +\infty \)) or negative infinity (\( -\infty \)). The end behavior is largely governed by the leading term of a polynomial.

• For instance, with the polynomial \( f(x) = -x^4 \), the leading term is \( -x^4 \), which directly affects its end behavior.
• Since \( -x^4 \) is a negative term and even-powered, it implies that as \( x \) goes to either infinity, the function \( f(x) \) will approach \( -\infty \).
• This consistent downward trend in the graph's tails highlights the end behavior dictated by the dominant (highest degree) term.

Recognizing the end behavior of a function, particularly polynomial ones, is essential, as it summarizes the ultimate trends and patterns in a graph, making it easier to predict how the function behaves over vast ranges of \( x \).