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Chapter 7: Problem 5

Evaluate \(\lim _{n \rightarrow \infty}\left(1+\frac{3}{n}\right)^{n}\)

Short Answer

Expert verified
The limit \(\lim_{n\to\infty} \left(1+\frac{3}{n}\right)^n = e^3\).

Step by step solution

01

Rewrite the expression with a substitution#

In order to manipulate the expression into a more convenient form, let's define a new variable \(x = \frac{3}{n}\). Then, we have \(n = \frac{3}{x}\). Now, rewrite the expression as: \(\lim _{x \rightarrow 0}\left(1+x\right)^{\left(\frac{3}{x}\right)}\). We can see that as \(n\) approaches infinity, \(x\) approaches 0.
02

Rewriting the power of the expression#

Recall that \(a^{bc} = (a^b)^c\). By using this rule, we can rewrite the expression as: \(\lim _{x \rightarrow 0}\left[\left(1+x\right)^{\left(\frac{1}{x}\right)}\right]^3\). Now, the expression inside the square brackets is reminiscent of the definition of the number \(e\).
03

Evaluate the limit#

We know that \(\lim _{x\rightarrow 0}\left(1+x\right)^{\frac{1}{x}} = e\). So our expression becomes: \[\lim _{x \rightarrow 0}\left[\left(1+x\right)^{\left(\frac{1}{x}\right)}\right]^3 = e^3.\] Therefore, the limit we were looking for is equal to \(e^3\).

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