find the limit of the following sequence or determine that the sequence diverges. {3^n / (3^n + 7^n)} select…

Explanation:

Step1: Divide numerator and denominator by $7^n$

[ \begin{align*} \lim_{n\rightarrow\infty}\frac{3^n}{3^n + 7^n}&=\lim_{n\rightarrow\infty}\frac{\frac{3^n}{7^n}}{\frac{3^n}{7^n}+\frac{7^n}{7^n}}\ &=\lim_{n\rightarrow\infty}\frac{(\frac{3}{7})^n}{(\frac{3}{7})^n + 1} \end{align*} ]

Step2: Evaluate the limit

Since $|\frac{3}{7}|<1$, we know that $\lim_{n\rightarrow\infty}(\frac{3}{7})^n = 0$. Then [ \begin{align*} \lim_{n\rightarrow\infty}\frac{(\frac{3}{7})^n}{(\frac{3}{7})^n + 1}&=\frac{\lim_{n\rightarrow\infty}(\frac{3}{7})^n}{\lim_{n\rightarrow\infty}(\frac{3}{7})^n+1}\ &=\frac{0}{0 + 1}=0 \end{align*} ]

Answer:

A. The limit of the sequence is $0$