- The function \(f(x) = x^7\) is a power function with an odd degree \(n=7\) and a positive leading coefficient \(a=1\).
- As \(x\) approaches positive infinity (\(x \rightarrow \infty\)), the positive values raised to the 7th power grow without bound: \(f(x) \rightarrow \infty\).
- As \(x\) approaches negative infinity (\(x \rightarrow -\infty\)), the negative values raised to the odd 7th power remain negative and their absolute values grow without bound: \(f(x) \rightarrow -\infty\).
- Final limits:
As \(x \rightarrow -\infty\), \(f(x) \rightarrow \boxed{-\infty}\)
As \(x \rightarrow \infty\), \(f(x) \rightarrow \boxed{\infty}\)
Explanation
- Function Analysis
The function given is \(f(x) = x^7\). This is a power function of the form \(f(x) = ax^n\), where \(a = 1\) and \(n = 7\). The exponent \(n=7\) is an odd positive integer, and the coefficient \(a=1\) is positive. These characteristics determine the end behavior, or "long-run behavior," of the function, which describes the output as \(x\) becomes very large in the positive or negative direction.
- Behavior as \(x \rightarrow \infty\)
To find the limit as \(x \rightarrow \infty\), consider substituting increasingly large positive values for \(x\). Since \(x\) is positive, \(x^7\) will also be positive. As the base \(x\) grows without bound, the power \(x^7\) grows exponentially faster, also without bound. Thus, \(\lim_{x \rightarrow \infty} x^7 = \infty\).
- Behavior as \(x \rightarrow -\infty\)
To find the limit as \(x \rightarrow -\infty\), consider substituting increasingly large negative values for \(x\). Because the exponent \(n=7\) is odd, raising a negative number to the power of 7 preserves the negative sign. Specifically, if \(x = -k\) (where \(k > 0\)), then \(x^7 = (-k)^7 = -k^7\). As \(x\) becomes more negative (i.e., \(k \rightarrow \infty\)), \(-k^7\) becomes more negative without bound. Thus, \(\lim_{x \rightarrow -\infty} x^7 = -\infty\).
- Final Conclusions and Results
Combining these results, we conclude that the function \(f(x) = x^7\) increases towards positive infinity as \(x\) approaches positive infinity, and decreases towards negative infinity as \(x\) approaches negative infinity.
As \(x \rightarrow -\infty\), \(f(x) \rightarrow -\infty\)
As \(x \rightarrow \infty\), \(f(x) \rightarrow \infty\)
Examples
Long-run behavior analysis is foundational in modeling physical systems. For example, consider an object accelerating under a force that scales with the cube or higher power of velocity; understanding that the force increases rapidly as velocity increases (the long-run behavior) helps engineers design systems that can handle extreme conditions or predict when a system might reach physical limits. This type of analysis ensures that mathematical models align with the expected performance of real-world mechanical or electrical components.