Euler's Number Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Euler's Number.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Euler's number e \approx 2.71828 is the unique base for which the exponential function e^x is its own derivative — the natural base for growth and decay.

The 'natural' base for growth—what you get from continuous compounding.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: e^x is its own derivative—it's the natural language of calculus.

Common stuck point: e is irrational and transcendental—its digits never repeat or end.

Sense of Study hint: Try computing (1 + 1/n)^n for n = 1, 10, 100, 1000 on a calculator and watch the values approach 2.718.

Worked Examples

Example 1

easy
Evaluate \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n and state what this limit defines.

Solution

  1. 1
    Substitute increasing values of n: for n=10, \left(1+\frac{1}{10}\right)^{10} \approx 2.5937; for n=100, \approx 2.7048; for n=1000, \approx 2.7169.
  2. 2
    Observe that as n \to \infty the expression approaches a fixed value that does not grow without bound.
  3. 3
    By definition, \lim_{n \to \infty}\left(1+\frac{1}{n}\right)^n = e \approx 2.71828\ldots, Euler's number.

Answer

e \approx 2.71828
Euler's number e is defined as this limit. It is irrational and transcendental, appearing naturally in continuous growth, calculus, and complex analysis.

Example 2

hard
Show that the derivative of f(x) = e^x is itself, i.e., f'(x) = e^x, using the limit definition of the derivative.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Using a calculator, compute e^2 to four decimal places. Then determine whether e^2 > 7.

Example 2

medium
A bank offers continuous compounding at an annual rate of 5\%. Using A = Pe^{rt}, find how much \1000 grows to after 10$ years.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

exponential function