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Chapter 7: Problem 2

Evaluate \(\lim _{n \rightarrow \infty} \frac{4 n-2}{7 n+6}\).

Short Answer

Expert verified
The limit of the given function as n approaches infinity is \(\lim _{n \rightarrow \infty} \frac{4 n-2}{7 n+6} = \frac{4}{7}\).

Step by step solution

01

Identify the highest power of n

In this case, the highest power of n is 1, as the numerator and the denominator are linear functions where each has n raised to the power 1.
02

Divide each term in the fraction by the highest power of n

Now we will divide each term in the numerator and the denominator by \(n^1\), which is just n. \[\frac{4n - 2}{7n + 6} \times \frac{1/n}{1/n} = \frac{4 - \frac{2}{n}}{7 + \frac{6}{n}}\]
03

Evaluate the limit as n approaches infinity

As n approaches infinity, the terms that contain n in the denominator will approach zero: \[\lim_{n \rightarrow \infty} \frac{4 - \frac{2}{n}}{7 + \frac{6}{n}} = \frac{4 - 0}{7 + 0} = \frac{4}{7}\] So, the limit of the given function as n approaches infinity is: \[\lim _{n \rightarrow \infty} \frac{4 n-2}{7 n+6} = \frac{4}{7}\]

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