Problem 20 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Analyze the Function

First, we identify the highest power of the exponential term in both the numerator and the denominator, which is \( e^{3x} \). The function is \( \frac{3 e^{2 x}}{2 e^{2 x}-e^{3 x}} \).

Step 2: Simplify the Function

Divide each term in the numerator and the denominator by \( e^{3x} \), which is the term with the highest power. This gives us \( \frac{\frac{3 e^{2x}}{e^{3x}}}{\frac{2 e^{2x}}{e^{3x}} - \frac{e^{3x}}{e^{3x}}} \).

Step 3: Simplify Each Term

Simplifying each term gives \( \frac{\frac{3}{e^{x}}}{\frac{2}{e^{x}} - 1} \).

Step 4: Evaluate Limits of Each Term

As \( x \to \infty \), \( \frac{3}{e^{x}} \to 0 \) and \( \frac{2}{e^{x}} \to 0 \). So, the simplified function becomes \( \frac{0}{0 - 1} = 0 \).

Step 5: Evaluate the Limit

Thus, the limit \( \lim _{x \rightarrow \infty} \frac{3 e^{2 x}}{2 e^{2 x}-e^{3 x}} \) evaluates to \( 0 \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where variables appear as exponents. The general form is:

• An exponential function can be represented as \( f(x) = a \cdot e^{bx} \), where \( a \) and \( b \) are constants, and \( e \) is the base of the natural logarithm, approximately 2.718.
• The variable \( x \) represents the exponent.

A key feature of exponential functions is their rapid growth or decay. In this exercise, we observe the function \( \frac{3 e^{2 x}}{2 e^{2 x} - e^{3 x}} \).
The numerator and denominator contain exponential expressions. Notice that \( e^{3x} \) grows faster than both \( e^{2x} \) and any constant coefficients as \( x \) approaches infinity. This characteristic is crucial when analyzing and simplifying such functions to evaluate limits.
Understanding the behavior of exponential functions helps us in simplifying expressions by isolating the dominant terms, thereby allowing for straightforward calculations.

Limit Evaluation

In calculus, evaluating the limit of a function as a variable approaches a specific value helps us understand its behavior. This exercise focuses on evaluating a limit as \( x \to \infty \). The concept involves determining what value, if any, a function approaches or converges to as the variable becomes very large.For this problem:

• Begin by identifying the highest exponential power in the expression, which is \( e^{3x} \).
• To simplify the evaluation process, divide each term by \( e^{3x} \), the largest exponential term in the function.

This technique reduces complicated terms and reveals their behavior as \( x \to \infty \). The simplified expression becomes \( \frac{\frac{3}{e^x}}{\frac{2}{e^x} - 1} \).
As \( x \to \infty \), \( \frac{3}{e^x} \) and \( \frac{2}{e^x} \) both approach zero, simplifying the limit evaluation to \( \frac{0}{0 - 1} = 0 \).
Limit evaluation plays a significant role in calculus, providing insights into function trends over large scales.

Mathematical Simplification

Mathematical simplification is the process of transforming complex expressions into simpler, more manageable forms while retaining their original values. This is fundamental in calculus, particularly for easy evaluation of limits, derivatives, and integrals.In our problem, the simplification process involved:

• Identifying the term with the highest growth rate, \( e^{3x} \).
• Dividing each component of the function by this term to reduce complexity.

By performing these steps, the expression was translated from a cumbersome format into \( \frac{\frac{3}{e^x}}{\frac{2}{e^x} - 1} \).
This transformation was vital because it allowed each part to be evaluated separately as \( x \to \infty \), leading to simplified terms such as \( \frac{3}{e^x} \) and \( \frac{2}{e^x} \), both tending to zero. In doing so, it turned the original complex function into a simple fraction, \( \frac{0}{-1} \), evaluating the limit to 0.
Simplification aids in revealing underlying trends and behaviors of functions, making limit evaluations and other calculus operations more accessible.