Problem 14 - Calculus

Question

In Exercises \(10-15,\) give \(\lim _{x \rightarrow-\infty} f(x)\) and \(\lim _{x \rightarrow+\infty} f(x)\). $$f(x)=8 x^{-3}$$

Step-by-Step Solution

Answer

\( \lim_{x \rightarrow -\infty} f(x) = 0 \) and \( \lim_{x \rightarrow +\infty} f(x) = 0 \).

Step 1: Understand the Function

The function given is \( f(x) = 8x^{-3} \). This is a polynomial function where \( x \) has a negative exponent of 3. It can be rewritten as \( f(x) = \frac{8}{x^3} \).

Step 2: Analyze the Behavior as \( x \) Approaches \(-\infty\)

As \( x \) approaches \(-\infty\), \( x^3 \) becomes very negative and large in magnitude, which means \( \frac{1}{x^3} \) becomes increasingly negative. Consequently, \( f(x) = 8 \times \frac{1}{x^3} \) approaches 0 from the negative side. Thus, \( \lim_{x \rightarrow -\infty} f(x) = 0 \).

Step 3: Analyze the Behavior as \( x \) Approaches \(+\infty\)

As \( x \) approaches \(+\infty\), \( x^3 \) becomes very positive and large in magnitude, so \( \frac{1}{x^3} \) becomes increasingly positive but small in magnitude. Therefore, \( f(x) = 8 \times \frac{1}{x^3} \) approaches 0 from the positive side. Thus, \( \lim_{x \rightarrow +\infty} f(x) = 0 \).

Step 4: Write the Conclusion

Both limits as \( x \rightarrow -\infty \) and \( x \rightarrow +\infty \) are 0. Therefore, the behavior of the function at both infinities is similar.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of a variable. In the function given, \( f(x) = 8x^{-3} \), we see a special form of **polynomial function** with a negative exponent. Typically, polynomial functions consist of terms like \( ax^n \), where \( a \) is a coefficient and \( n \) is a non-negative integer. However, in this function, the negative exponent makes it an interesting case to study.

The function can be rewritten as \( f(x) = \frac{8}{x^3} \) to make it more recognizable. It represents a form of inverse variation, typical for cases with negative powers. Understanding such transformations makes it easier to interpret behaviors, such as limits and asymptotes, which are central when discussing limits in calculus.

Behavior at Infinity

In calculus, the **behavior of functions at infinity** allows us to predict how a function behaves as the independent variable grows positively or negatively large. Analyzing the behavior of \( f(x) = 8x^{-3} \), we observe that:

• As \( x \to -\infty \), the term \( x^{-3} \) results in \( \frac{1}{x^3} \) becoming a very small negative value due to the cubic function's nature. Thus, \( f(x) \) tends to \( 0 \), but from the negative side.


• Conversely, as \( x \to +\infty \), \( x^{3} \) increases positively and \( \frac{1}{x^3} \) becomes a tiny positive value, driving the function \( f(x) \) towards \( 0 \) again, but from the positive side.

This demonstrates that as \( x \) moves towards positive or negative infinity, the function \( f(x) = 8x^{-3} \) approaches 0, indicating a sort of horizontal asymptotic behavior.

Negative Exponents

A negative exponent, like in \( f(x) = 8x^{-3} \), indicates a reciprocal relationship. Understanding this helps one in rewriting terms and comprehending their behavior in larger calculus contexts.

• When you see \( x^{-n} \), it means \( \frac{1}{x^n} \). It transforms the power into a division, presenting the expression as a fraction.


• Negative exponents diminish the value of a term as the variable becomes large in absolute value. Thus, they are crucial in analyzing limits, especially when discussing behavioral tendencies at infinity.

Overall, negative exponents provide a powerful tool in exploring functions' limits and predicting behaviors at extreme ranges of their variables.