Limiting Cases Math Example 4

Follow the full solution, then compare it with the other examples linked below.

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The binomial

(1+x/n)^n

has a famous limiting case. Evaluate for

n=1, 2, 10, 100

with

x=1

and identify the limit.

1
$n=1$ : $(1+1)^1 = 2$ .
2
$n=2$ : $(1+0.5)^2 = 1.5^2 = 2.25$ .
3
$n=10$ : $(1.1)^{10} \approx 2.5937$ .
4
$n=100$ : $(1.01)^{100} \approx 2.7048$ .
5
The limit as $n\to\infty$ is $e \approx 2.71828\ldots$

Answer

\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = e \approx 2.71828

This sequence of limiting cases defines Euler's number

e

. Each step shows the value creeping toward

e

from below, illustrating that

e

is a fundamental limiting case of exponential growth.

About Limiting Cases

Extreme values of a parameter (approaching zero, infinity, or a critical threshold) used to check formulas and reveal simplified behavior.

The formula for the sum of a geometric series [formula] ([formula]). Find the limiting case as [form

The compound interest formula is [formula]. Analyse the limiting cases [formula] (annual), [formula]

For the area of a regular [formula]-gon inscribed in a circle of radius [formula]: [formula]. What d