The Number e

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The number $e$ is defined by:

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

The number $e$ is a transcendental number which is approximately equal to $2.718281828 \dots$ The substitution $u = \frac{1}{n}$ where $u = \frac{1}{n} \to 0$ as $n \to \pm \infty,$ leads to another definition for $e :$

e = lim_{u \to 0} {(1 + u)}^{\frac{1}{u}} .

Here we meet with power expressions, in which the base and power approach to a certain number $a$ (or to infinity). In many cases such types of limits can be calculated by taking logarithm of the function.

Solved Problems

Click or tap a problem to see the solution.

Example 1

Calculate the limit $lim_{n \to \infty} {(1 + \frac{1}{n})}^{n + 5} .$

Example 2

Find the limit $lim_{x \to \infty} {(1 + \frac{1}{x})}^{3 x} .$

Example 3

Calculate the limit $lim_{x \to \infty} {(1 + \frac{6}{x})}^{x} .$

Example 4

Find the limit $lim_{x \to 0} \sqrt[x]{1 + 3 x} .$

Example 1.

Calculate the limit $lim_{n \to \infty} {(1 + \frac{1}{n})}^{n + 5} .$

Solution.

lim_{n \to \infty} {(1 + \frac{1}{n})}^{n + 5} = lim_{n \to \infty} [{(1 + \frac{1}{n})}^{n} {(1 + \frac{1}{n})}^{5}] = lim_{n \to \infty} {(1 + \frac{1}{n})}^{n} \cdot lim_{n \to \infty} {(1 + \frac{1}{n})}^{5} = e \cdot 1 = e .

Example 2.

Find the limit $lim_{x \to \infty} {(1 + \frac{1}{x})}^{3 x} .$

Solution.

By the product rule for limits, we obtain

lim_{x \to \infty} {(1 + \frac{1}{x})}^{3 x} = lim_{x \to \infty} {(1 + \frac{1}{x})}^{x} \cdot lim_{x \to \infty} {(1 + \frac{1}{x})}^{x} \cdot lim_{x \to \infty} {(1 + \frac{1}{x})}^{x} = e \cdot e \cdot e = e^{3} .

Example 3.

Calculate the limit $lim_{x \to \infty} {(1 + \frac{6}{x})}^{x} .$

Solution.

Substituting $\frac{6}{x} = \frac{1}{y},$ so that $x = 6 y$ and $y \to \infty$ as $x \to \infty,$ we obtain

lim_{x \to \infty} {(1 + \frac{6}{x})}^{x} = lim_{y \to \infty} {(1 + \frac{1}{y})}^{6 y} = lim_{y \to \infty} {[{(1 + \frac{1}{y})}^{y}]}^{6} = {[lim_{y \to \infty} {(1 + \frac{1}{y})}^{y}]}^{6} = e^{6} .

Example 4.

Find the limit $lim_{x \to 0} \sqrt[x]{1 + 3 x} .$

Solution.

lim_{x \to 0} \sqrt[x]{1 + 3 x} = lim_{x \to 0} {(1 + 3 x)}^{\frac{1}{x}} = lim_{3 x \to 0} {(1 + 3 x)}^{\frac{1}{3 x} \cdot 3} = lim_{3 x \to 0} {[{(1 + 3 x)}^{\frac{1}{3 x}}]}^{3} = {[lim_{3 x \to 0} {(1 + 3 x)}^{\frac{1}{3 x}}]}^{3} = e^{3} .