Definition of the exponential function for MATH 140

Exam Relevance for MATH 140

Likelihood of appearing: Essential

The definition of e is conceptual background in MATH 140. Underlies exponential derivative formulas.

Lesson

What is the Definition of the Exponential Function?

The number $e \approx 2.71828...$ isn't just a random constant—it emerges naturally from a specific limit. Understanding this limit definition is essential because many exam problems are impossible to solve without recognizing this pattern.

The Fundamental Limit:

$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$$

This says: as $n$ gets larger and larger, the expression $\left(1 + \frac{1}{n}\right)^n$ approaches $e$.

🔑 Why does this matter? Many limits on exams are disguised versions of this definition. If you don't recognize the pattern, the limit appears unsolvable—there's no algebraic trick that will help you.

The General Form

The definition generalizes to:

$$\lim_{n \to \infty} \left(1 + \frac{a}{n}\right)^{bn} = e^{ab}$$

Key insight: The exponent times the fraction's numerator gives you the power of $e$.

When you see a limit of the form $(1 + \frac{\text{something}}{\text{variable}})^{\text{power involving variable}}$, think: "This is the definition of $e$!"

How to Recognize These Problems

Look for limits with this structure:

Base of the form $(1 + \frac{\text{const}}{x})$ or $(1 + \frac{\text{const}}{n})$
Exponent involving $x$ or $n$
The variable goes to $\infty$

Pattern matching:

$$\left(1 + \frac{a}{x}\right)^{bx} \to e^{ab}$$

The answer is always $e$ raised to (numerator of fraction) $\times$ (coefficient on $x$ in exponent).

Example 1: Basic Form

Problem: Evaluate $\displaystyle\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$

This is exactly the definition of $e$.

$$\boxed{e}$$

Example 2: With a Constant in the Numerator

Problem: Evaluate $\displaystyle\lim_{x \to \infty} \left(1 + \frac{4}{x}\right)^{3x}$

Step 1: Identify the pattern.

This has the form $\left(1 + \frac{a}{x}\right)^{bx}$ where:

$a = 4$ (numerator of the fraction)
$b = 3$ (coefficient of $x$ in the exponent)

Step 2: Apply the formula.

$$\lim_{x \to \infty} \left(1 + \frac{a}{x}\right)^{bx} = e^{ab} = e^{4 \cdot 3} = \boxed{e^{12}}$$

Example 3: Proving a Limit Equals $\frac{1}{e}$

Problem: Prove that $\displaystyle\lim_{n \to \infty} \left(\frac{n}{n+1}\right)^n = \frac{1}{e}$

Step 1: Rewrite to match the standard form.

$$\frac{n}{n+1} = \frac{1}{\frac{n+1}{n}} = \frac{1}{1 + \frac{1}{n}}$$

So: $$\left(\frac{n}{n+1}\right)^n = \left(\frac{1}{1 + \frac{1}{n}}\right)^n = \frac{1}{\left(1 + \frac{1}{n}\right)^n}$$

Step 2: Take the limit.

$$\lim_{n \to \infty} \frac{1}{\left(1 + \frac{1}{n}\right)^n} = \frac{1}{e}$$

$$\boxed{\frac{1}{e}}$$

Example 4: Coefficient Doesn't Match

Problem: Evaluate $\displaystyle\lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^{x}$

Step 1: Identify what we have.

We have $\left(1 + \frac{2}{x}\right)^{x}$, which is $\left(1 + \frac{a}{x}\right)^{bx}$ with $a = 2$ and $b = 1$.

Step 2: Apply the formula.

$$e^{ab} = e^{2 \cdot 1} = \boxed{e^2}$$

Example 5: Manipulation Required

Problem: Evaluate $\displaystyle\lim_{n \to \infty} \left(1 + \frac{3}{2n}\right)^{n}$

Step 1: The fraction has $2n$ in the denominator but the exponent is just $n$.

Let's rewrite to use the standard form. We want the denominator and exponent to match.

Step 2: Think of it as: $$\left(1 + \frac{3}{2n}\right)^{n} = \left[\left(1 + \frac{3}{2n}\right)^{2n}\right]^{1/2}$$

Now the inner expression has matching $2n$ in denominator and exponent.

Step 3: The inner limit: $$\lim_{n \to \infty} \left(1 + \frac{3}{2n}\right)^{2n} = e^3$$

Step 4: Apply the outer exponent: $$(e^3)^{1/2} = e^{3/2} = \boxed{e^{3/2}}$$

Alternative method: Use $a = \frac{3}{2}$ and $b = 1$ (thinking of $\frac{3}{2n}$ as $\frac{3/2}{n}$): $$e^{ab} = e^{(3/2)(1)} = e^{3/2}$$

Example 6: Variable Substitution

Problem: Evaluate $\displaystyle\lim_{x \to 0^+} (1 + x)^{1/x}$

Step 1: This looks different—$x \to 0$, not $x \to \infty$.

Let $n = \frac{1}{x}$. As $x \to 0^+$, we have $n \to \infty$.

Step 2: Substitute: $$(1 + x)^{1/x} = \left(1 + \frac{1}{n}\right)^{n}$$

Step 3: This is exactly the definition of $e$!

$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n} = \boxed{e}$$

Common Mistakes and Misunderstandings

❌ Mistake: Trying to use L'Hôpital's Rule or other techniques

Wrong: Attempting to rewrite and differentiate, or using algebraic manipulation to "solve" the limit.

Why it's wrong: This limit is a definition—it cannot be derived from simpler principles in Calc 1. You must recognize the pattern and apply the definition directly.

Correct: Recognize the form $\left(1 + \frac{a}{n}\right)^{bn} \to e^{ab}$ and use it.

❌ Mistake: Thinking $(1 + \frac{1}{n})^n \to 1$ because $\frac{1}{n} \to 0$

Wrong: "As $n \to \infty$, $\frac{1}{n} \to 0$, so $1 + \frac{1}{n} \to 1$, and $1^n = 1$."

Why it's wrong: This is a $1^\infty$ indeterminate form! The base approaches $1$ while the exponent approaches $\infty$—these compete, and the result is neither $1$ nor $\infty$, but $e$.

Correct: $\displaystyle\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e \approx 2.718...$

❌ Mistake: Forgetting to match the denominator and exponent

Wrong: For $\displaystyle\lim_{n \to \infty} \left(1 + \frac{5}{3n}\right)^{n}$, saying the answer is $e^5$.

Why it's wrong: The denominator is $3n$ but the exponent is $n$—they don't match! You need to account for this.

Correct: Rewrite as $\left[\left(1 + \frac{5}{3n}\right)^{3n}\right]^{1/3} \to (e^5)^{1/3} = e^{5/3}$ Or use: $a = \frac{5}{3}$, $b = 1$, so $e^{ab} = e^{5/3}$.

Quick Reference

Limit Form	Result
$\displaystyle\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$	$e$
$\displaystyle\lim_{n \to \infty} \left(1 + \frac{a}{n}\right)^{bn}$	$e^{ab}$
$\displaystyle\lim_{x \to 0} (1 + x)^{1/x}$	$e$
$\displaystyle\lim_{n \to \infty} \left(\frac{n}{n+1}\right)^n$	$\frac{1}{e}$

Formulas & Reference

Definition of e

\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e

The fundamental limit definition of Euler's number e ≈ 2.71828. This is a definition, not something you derive—you must recognize and apply it directly.