Problem 46 - Calculus for Biology and Medicine

Question

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow \infty}\left(1+\frac{5}{x}\right)^{x} $$

Step-by-Step Solution

Answer

The limit is \( e^5 \).

Step 1: Understand the Limit Form

We are given the limit \( \lim _{x \rightarrow \infty}\left(1+\frac{5}{x}\right)^{x} \). Notice that this expression is in the form similar to \( \left(1+\frac{1}{n}\right)^{n} \) which converges to \( e \) as \( n \to \infty \). We suspect that this limit is also related to the exponential function \( e^m \), where \( m \) is some constant.

Step 2: Use the Exponential Limit Definition

We can write the expression in the following form: \[ \left(1+\frac{5}{x}\right)^{x} = \exp\left(x \ln \left(1+\frac{5}{x}\right)\right) \]. This makes use of the identity \( a^b = \exp(b \ln a) \). The goal now is to find the limit of the exponent \( x \ln\left(1+\frac{5}{x}\right) \) as \( x \rightarrow \infty \).

Step 3: Simplify the Exponent

Using the fact that \( \ln(1+u) \approx u \) as \( u \to 0 \), we approximate: \[ \ln\left(1+\frac{5}{x}\right) \approx \frac{5}{x} \]. Thus, the exponent becomes \( x \cdot \frac{5}{x} = 5 \).

Step 4: Conclude the Limit

Given the simplification, we have: \[ x \ln\left(1+\frac{5}{x}\right) \approx 5 \] as \( x \to \infty \). Therefore, the original limit \( \exp(x \ln(1+\frac{5}{x})) \) simplifies to \( \exp(5) \).

Step 5: State the Final Answer

The limit \( \lim _{x \rightarrow \infty}\left(1+\frac{5}{x}\right)^{x} \) evaluates to \( e^{5} \).

Key Concepts

Exponential Functions

Exponential functions are fundamental in mathematics, particularly in calculus. These functions have the form \( f(x) = a^x \), where \( a \) is a constant known as the base, and \( x \) is the exponent. A common special case is when the base \( a \) is Euler's number \( e \), approximately 2.718. This leads to the natural exponential function written as \( f(x) = e^x \).

• Exponential functions are defined for all real numbers and are continuous and differentiable everywhere.
• They model a wide range of real-world phenomena, such as population growth, radioactive decay, and compound interest.
• The unique property of exponential growth is that the rate of growth is proportional to the current quantity present.

In the context of limits, exponential growth can show intriguing behaviors as it is highly responsive to changes in the exponent, especially as the exponent approaches infinity.

l'Hôpital's Rule

l'Hôpital's Rule is a powerful tool in calculus used to evaluate limits that present indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Named after the French mathematician Guillaume de l'Hôpital, the rule states that if \( \lim_{x\to c} f(x) = 0 \) and \( \lim_{x\to c} g(x) = 0 \) or both approach infinity, and the derivatives \( f'(x) \) and \( g'(x) \) exist near \( c \), then:\[ \lim_{x\to c} \frac{f(x)}{g(x)} = \lim_{x\to c} \frac{f'(x)}{g'(x)} \]if the limit on the right side exists.

• It's important to apply l'Hôpital's Rule only when the original limit is of an indeterminate form.
• Multiple applications of l'Hôpital's Rule might be needed if the form remains indeterminate after the first application.

While l'Hôpital's Rule is immensely useful, it's not necessary for the original exercise. Here, recognizing properties of exponential functions and using approximations suffices to find the limit.

Natural Logarithms

Natural logarithms are logarithms to the base \( e \), referred to as \( \ln(x) \). They have important properties that make them widely used in both pure and applied mathematics.

• They are the inverse of an exponential function, meaning \( \ln(e^x) = x \) and \( e^{\ln(x)} = x \) for \( x > 0 \).
• Natural logarithms often simplify differentiation and integration problems, especially when dealing with exponential functions.

In the given problem, the transformation \( \left(1+\frac{5}{x}\right)^x \) is expressed as an exponential using the natural logarithm: \( \exp(x \ln(1+\frac{5}{x})) \). This step simplifies analyzing the behavior of the expression as \( x \to \infty \). Recognizing how natural logarithms approximate \( \ln(1+u) \approx u \) when \( u \to 0 \) is crucial for evaluating such limits effectively.