Problem 23 - Calculus for Biology and Medicine

Question

Evaluate the limits. $$ \lim _{x \rightarrow \infty} \frac{2}{e^{x}(1+x)} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Limit Expression

We need to evaluate the limit \( \lim_{x \to \infty} \frac{2}{e^x(1+x)} \). As \( x \) approaches infinity, both \( e^x \) and \( 1+x \) grow very fast, making the denominator grow larger.

Step 2: Analyze the Growth of Denominator

Both \( e^x \) and \( 1+x \) increase significantly as \( x \) approaches infinity. Specifically, \( e^x \) (the exponential function) increases much faster than \( 1+x \) (the linear function).

Step 3: Simplify Exponential Dominance

Given that exponential functions grow faster than linear functions, the term \( e^x \cdot (1+x) \) in the denominator will dominate the fraction as \( x \to \infty \). Therefore, \( \frac{2}{e^x(1+x)} \) should approach zero.

Step 4: Conclude the Limit

Thus, due to the rapid growth of the denominator, the value of the expression \( \frac{2}{e^x(1+x)} \) approaches zero as \( x \to \infty \). Therefore, the limit evaluates to:

Key Concepts

Exponential Growth

Exponential growth is a powerful concept in mathematics, especially when evaluating limits. It refers to an increase in quantity at a consistent rate, multiplying over time. For example, the function \( e^x \), where \( e \) is the base of the natural logarithm, is characterized by rapid growth as \( x \) becomes larger.
This type of growth is faster than any polynomial or linear growth, meaning that when \( x \) moves toward infinity, exponential functions will exceed other types of functions.

• **Exponential Functions:** These are functions with variables in the exponent, such as \( e^x \).
• **Growth Rate Comparisons:** For large values of \( x \), an exponential function's value far outpaces functions like \( 1+x \).

In our limit problem, \( e^x(1+x) \), the exponential component \( e^x \) outgrows the linear function \( 1+x \) rapidly, playing a crucial role in determining the behavior of the denominator.

Denominator Analysis

Analyzing the denominator is crucial in evaluating limits. When the denominator of a fraction grows very large, the entire fraction itself tends to zero, provided that the numerator is fixed or grows at a slower rate.
For our function, we have the expression \( \frac{2}{e^x(1+x)} \), with the denominator \( e^x(1+x) \). As \( x \to \infty \), both components: exponential \( e^x \) and linear \( 1+x \), increase.

• **Dominance of Exponential Growth:** The exponential term \( e^x \) significantly impacts the denominator's growth due to its rapid increase compared to \( 1+x \).
• **Effect on the Fraction:** With a large denominator, the fraction's value \( \frac{2}{e^x(1+x)} \) approaches zero.

Denominator analysis highlights how the underlying functions' growth rates determine the entire expression's behavior.

Infinite Limits

Infinite limits explore behavior as variables grow without bound, typically moving toward infinity. This involves understanding how different mathematical expressions behave as inputs become infinitely large.
In our given problem, we're interested in \( \lim_{x \to \infty} \frac{2}{e^x(1+x)} \). The concept of infinite limits helps identify whether an expression approaches a finite value or a specific direction like zero.

• **Growth Comparison:** Exponential growth at \( e^x \) dominates, impacting the limit significantly as larger factors in the denominator cause the fraction to shrink towards zero.
• **Role of Constants:** The constant numerator \( 2 \) remains unchanged, while the ever-growing denominator ensures the fraction diminishes as \( x \to \infty \).

Understanding infinite limits gives clarity on why facets like exponential growth in the denominator cause the overall limit to resolve effectively as zero.