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

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

Question
Limits and continuity
asked 2020-12-16
Find the following complex limits. \(\lim_{n\rightarrow\infty}\frac{2n}{7n+ni}\)

Answers (1)

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 \(\lim_{n\rightarrow\infty}\frac{2n}{7n+ni}\)
\(\lim_{n\rightarrow\infty}\frac{2n}{7n+ni}=\lim_{n\rightarrow\infty}\frac{7-i}{25}\)
\(=\frac{7-i}{25}\)
\(=\frac{7}{25}-\frac{1}{25i}\)
0

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