Problem 33 - Calculus for Biology and Medicine

Question

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow+\infty}\left(e^{x}-x^{3}\right) $$

Step-by-Step Solution

Answer

The limit is \(+\infty\) as \(x\) approaches \(+\infty\).

Step 1: Analyze Function Components

Consider the function inside the limit \( e^x - x^3 \) as \( x \) approaches infinity. The function consists of an exponential function \( e^x \) and a polynomial \( x^3 \). Our task is to determine which component grows faster as \( x \rightarrow +\infty \).

Step 2: Compare Growth Rates

The exponential function \( e^x \) grows much faster than any polynomial as \( x \) approaches infinity. Thus, \( e^x \) becomes significantly larger than \( x^3 \), leading the behavior of the overall expression.

Step 3: Apply Limit Concept

Since \( e^x \) dominates \( x^3 \), the subtraction \( e^x - x^3 \) behaves similarly to \( e^x \) alone as \( x \rightarrow +\infty \). This is because the exponential growth of \( e^x \) overshadows the polynomial growth of \( x^3 \).

Step 4: Conclude the Limit

Hence, the limit \( \lim_{x \rightarrow +\infty}(e^x - x^3) = +\infty \), as the rapidly increasing \( e^x \) dictates the limit of the entire expression.

Key Concepts

Limit

A limit in calculus refers to the value that a function approaches as the input approaches some value. In this particular exercise, we are looking at what happens to the function as the variable \( x \) becomes extremely large, approaching infinity. Understanding limits is crucial because they help us analyze the behavior of functions at points that might not be easily evaluated with direct substitution.
For this problem, the limit is expressed as \( \lim _{x \rightarrow +\infty}(e^x - x^3) \). This notation tells us to look at the behavior of the expression \( e^x - x^3 \) as \( x \) becomes infinitely large. When evaluating such limits, we focus on the dominant term, the one that grows or declines the fastest, to determine how the overall expression behaves.

• If the dominant term grows without bounds, the limit typically results in infinity.
• If it approaches a finite number, then the limit is that number.

In this problem, determining the limit clarifies how each component contributes to the function's behavior as it approaches a vast boundary.

Exponential Function

Exponential functions are a key part of calculus and mathematics. The specific function \( e^x \) is an exponential function where the base \( e \) (approximately 2.718) raised to the power of \( x \). It’s important to note that exponential functions grow extremely fast, especially as \( x \) becomes larger. This fast growth is a critical characteristic that distinguishes exponential functions from other types of functions, such as polynomials or linear functions.
For exponential functions like \( e^x \):

• They increase rapidly as \( x \) increases.
• Even a small increase in \( x \) results in a large increase in the value of \( e^x \).

When comparing exponential functions to others, they often dominate the behavior as \( x \) approaches infinity because their rate of increase is much higher. This dominance is why, in our problem, the term \( e^x \) overshadows \( x^3 \) as \( x \to \infty \), leading the overall function \( e^x - x^3 \) towards infinity.

Polynomial Growth

Polynomial growth describes functions that are defined by polynomial equations, like \( x^3 \). These functions grow at a rate dictated by the highest power of \( x \), known as the degree of the polynomial. While polynomials do grow as \( x \) becomes large, they do not grow as quickly as exponential functions.
In the context of the problem:

• \( x^3 \) is a polynomial and grows relatively slowly compared to \( e^x \).
• As \( x \to \infty \), \( x^3 \) will increase, but its growth will be trailed by \( e^x \), which escalates rapidly.

Thus, although \( x^3 \) contributes to the overall expression, its influence diminishes in comparison to the exponential term. Consequently, when evaluating how the entire expression \( e^x - x^3 \) behaves as \( x \) gets large, the exponential component becomes the main focus, and the impact of the polynomial growth becomes negligible.