Problem 69 - Essential Calculus

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 \(\lim _{x \rightarrow \infty}\left[p(x) / e^{x}\right]=0\).

Step-by-Step Solution

Answer

The statement is true.

Step 1: 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.

Step 2: 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.

Step 3: Conclusion

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

Key Concepts

Polynomial Functions

Polynomial functions are a class of mathematical expressions built from variables and constants. They are combined using only addition, subtraction, multiplication, and non-negative integer exponents. These functions can have one or more terms. For example, a polynomial in the variable \( x \) could look like \( p(x) = 3x^3 + 2x^2 - x + 5 \). Here,

• The degree of the polynomial is the highest power of \( x \), which is 3 in this case.
• The coefficients are the numbers multiplying each term.

As the input \( x \) becomes very large, the term with the largest power dominates the behavior of the polynomial function. This means that even if you have multiple terms, the one with the highest degree will affect the limit behavior the most. Understanding this concept is crucial when evaluating the limit of polynomial functions as \( x \) approaches infinity.

Exponential Functions

Exponential functions are a powerful form of functions where the variable appears in the exponent. A typical exponential function looks like \( f(x) = a^x \), where \( a \) is a positive constant. One of the most common bases used in calculus is the number \( e \), which is approximately 2.718. For an exponential function \( e^x \),

• The function grows very rapidly as \( x \) increases.
• Unlike polynomial functions, exponential functions increase their rate of growth, which means they eventually surpass polynomials regardless of the polynomial's degree.

When analyzing limits, particularly as \( x \) approaches infinity, exponential functions dominate, leading many polynomial-exponential comparisons to reduce to zero in such contexts. This principle underpins the conclusion that \( \lim_{x \to \infty} [p(x) / e^x] = 0 \) for polynomial \( p(x) \).

Infinity Limits

Infinity limits explore the behavior of functions as they either grow very large or very small, heading towards infinity or negative infinity. When we say the limit of \( f(x) \) as \( x \to \infty \) is \( L \), it means that as \( x \) grows indefinitely, the function approaches a specific value \( L \). In calculus, understanding limits involving infinity helps us grasp the end behavior of various functions.
When comparing polynomial functions and exponential functions,

• It's important to note that exponential functions, like \( e^x \), grow faster than polynomial functions as \( x \to \infty \).
• Thus, the ratio \( \frac{p(x)}{e^x} \) becomes very small and tends towards zero because the denominator \( e^x \) increases much faster than any polynomial numerator \( p(x) \).

This behavior is why the statement in the exercise holds true. Exploring how functions behave as they approach positive and negative infinity helps broaden our understanding of their growth patterns and limits.