Question

True or False? In Exercises 67-70, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(p(x)\) is a polynomial, then \(\lim _{x \rightarrow \infty}\left[p(x) / e^{x}\right]=0\).

Short answer

The statement is true.

Step-by-step solution

Understand the Statement

The statement is claiming that for any polynomial function \(p(x)\), as \(x\) approaches infinity, the ratio of the polynomial to the exponential function \(e^x\) will approach 0. Essentially, this means as \(x\) gets larger and larger, the exponential function will grow faster than any polynomial, making the ratio approach 0.

Analyze the Properties of Exponential and Polynomial Functions

Exponential functions grow at a much faster rate than polynomial functions as \(x\) approaches infinity. No matter how large the degree of the polynomial, the exponential function will eventually surpass it. Thus, the ratio \(p(x) / e^x\) should approach 0 as \(x\) approaches infinity.

Conclusion

The statement is consistent with the properties of polynomial and exponential functions, so it is deemed to be true.