Problem 15 - Calculus for Biology and Medicine

Question

Evaluate the limits in problems. $$ \lim _{x \rightarrow \infty} \frac{2 e^{x}}{e^{x}+3} $$

Step-by-Step Solution

Answer

The limit is 2.

Step 1: Identify the Form

The given expression is \( \lim _{x \rightarrow \infty} \frac{2e^{x}}{e^{x}+3} \). Here, \( e^x \) is an exponential function that grows very quickly as \( x \) approaches infinity. Therefore, both the numerator and denominator have \( e^x \) as a common term.

Step 2: Simplify by Factoring Out the Dominant Term

Factor out \( e^x \) from both the numerator and denominator: \( \frac{2e^{x}}{e^{x}+3} = \frac{e^{x}(2)}{e^{x}(1 + \frac{3}{e^{x}})} \). This simplifies to \( \frac{2}{1 + \frac{3}{e^{x}}} \).

Step 3: Evaluate the Limit

As \( x \rightarrow \infty \), the term \( \frac{3}{e^{x}} \) approaches 0 because \( e^x \) becomes very large. Therefore, the expression becomes \( \frac{2}{1+0} = 2 \).

Step 4: Conclude the Solution

Thus, the limit of the expression as \( x \) approaches infinity is 2. Write down the final evaluated limit as \( \lim _{x \rightarrow \infty} \frac{2 e^{x}}{e^{x}+3} = 2 \).

Key Concepts

Understanding Exponential Functions

Exponential functions are powerful tools in mathematics, often used to model real-world phenomena such as population growth and radioactive decay. They are defined by the formula \( f(x) = a e^{bx} \), where \( a \) and \( b \) are constants, and \( e \) is the base of natural logarithms, approximately equal to 2.718. This natural exponential function, denoted as \( e^x \), increases very rapidly as \( x \) becomes larger.

Key properties of exponential functions include:

• Their growth rate. For positive values of \( b \), exponential functions grow faster than polynomial functions, which is why they dominate in terms of size as \( x \) approaches infinity.
• The vertical asymptote behavior. For exponential functions \( e^{-x} \), as \( x \) approaches infinity, the function value trends towards zero, showing a horizontal asymptote.

These characteristics make exponential functions crucial for understanding limits, especially those involving infinity.

Infinity Behavior in Calculus

In the realm of calculus, dealing with expressions as they approach infinity is fundamental. When we say "\( x \rightarrow \infty \)", we are considering the behavior of functions as \( x \) grows without bound. This perspective often involves noting which terms in a function grow faster or slower compared to others.

For instance, in the expression \( \frac{2 e^{x}}{e^{x}+3} \), both the numerator and the dominant part of the denominator (\( e^x \)) grow exponentially. This means they increase at the same rate as \( x \) heads towards infinity, simplifying limit evaluations.

Understanding how functions behave at infinity involves recognizing that terms with slower growth rates, like constants or polynomials (e.g., the '+3' in the denominator), become negligible, simplifying calculations.

This concept is pivotal in calculus because it aids in determining the long-term behavior and end-behavior models for functions.

Exploring Limit Evaluation Techniques

Limit evaluation techniques are vital for simplifying expressions, especially those involving a variable approaching infinity. For evaluating the limit \( \lim _{x \rightarrow } \frac{2 e^{x}}{e^{x}+3} \), the different growth rates help determine which terms in a function can be simplified or ignored as they become insignificant at infinity.

A common technique is to factor out the highest-power term in the numerator and denominator, thereby simplifying the expression. In this case, \( e^x \) was the dominant term. By factoring \( e^x \) out of the numerator and denominator, the expression was reduced to \( \frac{2}{1 + \frac{3}{e^{x}}} \) as the \( e^x \)'s cancelled each other out. This allows the smaller terms to vanish as \( x \rightarrow \infty \).

Additionally, L'Hôpital's Rule is often applied when an indeterminate form such as \( \frac{0}{0} \) or \( \frac{}{} \) is encountered. However, in this instance, simple factorization and substitution were sufficient. Mastering these techniques helps in evaluating complex limits and understanding the ultimate behavior of a function.