# Find the following complex limits. lim_{nrightarrowinfty}frac{2n}{7n+ni}

Question
Limits and continuity
Find the following complex limits. $$\lim_{n\rightarrow\infty}\frac{2n}{7n+ni}$$

2020-12-17
$$\lim_{n\rightarrow\infty}\frac{2n}{7n+ni}$$
$$\frac{2n}{7n+ni}=\frac{\frac{2n}{n}}{\frac{7n+ni}{n}}$$
$$=\frac{2}{7+i}$$
Rationalize $$\frac{2}{7+i}$$ as
$$\frac{2}{7+i}=\frac{2}{7+i}\cdot\frac{7-i}{7-i}$$
$$=\frac{2(7-i)}{49-i^2}$$
$$=\frac{2(7-i)}{49+1}$$
$$=\frac{2(7-i)}{50}$$
$$=\frac{7-i}{25}$$
Substitute $$\frac{2n}{7n+ni}=\frac{7-i}{25}$$ in $$\lim_{n\rightarrow\infty}\frac{2n}{7n+ni}$$
$$\lim_{n\rightarrow\infty}\frac{2n}{7n+ni}=\lim_{n\rightarrow\infty}\frac{7-i}{25}$$
$$=\frac{7-i}{25}$$
$$=\frac{7}{25}-\frac{1}{25i}$$

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