Problem 45 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is \( e^3 \).

Step 1: Recognize the Form

The expression \( \left(1+\frac{3}{x}\right)^{x} \) resembles the limit definition of the exponential function. As \( x \rightarrow \infty \), this expression is similar to \( \left(1+\frac{k}{n}\right)^{n} \rightarrow e^{k} \). Identify \( k = 3 \) here.

Step 2: Rewrite Using Known Limit Form

Recognize that \( \lim_{x \rightarrow \infty} \left(1 + \frac{3}{x}\right)^x \) can be associated with \( e \), since it is in the form \( \left(1+\frac{k}{n}\right)^{n} \rightarrow e^{k} \). Rewrite the limit as \( e^{k} \) where \( k = 3 \).

Step 3: Apply the Exponential Limit Definition

Using \( e^{3} = \lim_{x \rightarrow \infty} \left(1 + \frac{3}{x}\right)^x \), we explicitly write it utilizing the fact that such expressions are equivalently \( e \) raised to the power of the constant shared in the fraction.

Step 4: Conclude the Evaluation

Since we've rewritten the expression into a standard limit form, we conclude that the original limit evaluates to \( e^3 \), using the property \( \left(1+\frac{k}{n}\right)^{n} \rightarrow e^{k} \) as \( n \) approaches infinity.

Key Concepts

Exponential Function

The exponential function is one of the most fascinating and widely used functions in mathematics and real-world applications. When we talk about exponential functions, we often refer to expressions involving the constant base "e", approximately equal to 2.718.

• Exponential functions have the form \(e^x\), where \(e\) is Euler's number.
• These functions exhibit rapid growth or decay, making them crucial in fields like biology, finance, and physics.

In the context of limits, exponential functions come into play when expressions approach special forms, enabling them to be transformed into something more recognizable. For example, expressions of the form \( (1 + \frac{1}{n})^n \) converge to \(e\) as \(n\) approaches infinity. In this exercise, we examined the expression \( \left(1+\frac{3}{x}\right)^{x} \) as \(x \rightarrow \infty\) and found it analogous to \( (1+\frac{k}{n})^n \). This means it evaluates to \(e^k\). So here, since \(k = 3\), the expression converges to \(e^3\). Understanding these connections between numbers and exponential functions is key to mastering calculus limits.

Limit Definition

Limits help us understand the behavior of functions as they approach specific points or infinity. In calculus, taking a limit allows us to explore how a function behaves as its input becomes very large or very small. When we look at a limit like \( \lim _{x \rightarrow \infty}\left(1+\frac{3}{x}\right)^{x} \), we aim to see what the function's value approaches when \(x\) becomes endlessly large.

• The expression given is a classic example illustrating how limits connect to the exponential function.

Using limit properties, we can often rewrite complex expressions into simpler, well-known forms. This expression, for example, takes a form that is equivalent to \(e^k\) due to a fundamental limit definition identified by Euler. Recognizing limits and rewriting them using known forms is a powerful technique to simplify and solve complex calculus problems.

l'Hôpital's Rule

l'Hôpital's rule is a tool in calculus used to find limits of indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). When an expression consists of such forms, evaluating directly can be challenging. However, in this limit problem, l'Hôpital's rule is not used. Instead, the recognition of the expression's form as a case of an exponential limit simplifies the process.

• l'Hôpital's rule guides us through differentiation when typical limit techniques fail or require simplification.
• By differentiating the numerator and denominator separately, we can re-evaluate the limit under simpler conditions.

Though this rule is not needed here, it's essential to learning about limits as it can directly tackle challenging expressions by breaking them into more manageable parts. Knowing when and how to apply l'Hôpital's rule is crucial when dealing with calculus limits, especially those involving indeterminate forms.