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Chapter 9: Problem 38

Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. $$a_{n}=\frac{3^{n}}{3^{n}+4^{n}}$$

Short Answer

Expert verified
Answer: The limit of the sequence is 0.

Step by step solution

01

Write the given sequence

We are given the sequence: $$a_{n}=\frac{3^{n}}{3^{n}+4^{n}}$$
02

Divide by the largest exponent

As n approaches infinity, the largest exponent is \(4^n\), so we divide both the numerator and the denominator by \(4^n\): $$\lim_{n\to\infty} a_{n} = \lim_{n\to\infty} \frac{\frac{3^n}{4^n}}{\frac{3^n}{4^n}+\frac{4^n}{4^n}}$$
03

Simplify the expression

Now rewrite the expression using the property \(a^{m}/a^{n}=a^{m-n}\) for the exponent: $$\lim_{n\to\infty} a_{n} = \lim_{n\to\infty} \frac{(3/4)^n}{(3/4)^n+1}$$
04

Determine the limit

As n approaches infinity, the value of \((3/4)^n\) approaches 0, due to the fact that the base, \(\frac{3}{4}\), is between 0 and 1: $$\lim_{n\to\infty} a_{n}=\frac{0}{0+1}=0$$
05

Verify the result with a graphing utility

To confirm the result, graph the sequence using a graphing utility. You should observe that, as the value of n increases, the value of the sequence approaches 0. Thus, the limit of the sequence is 0.

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