Find the following complex limits. lim_{nrightarrowinfty}frac{1}{3+ni}

Question
Limits and continuity
asked 2021-02-03
Find the following complex limits. \(\lim_{n\rightarrow\infty}\frac{1}{3+ni}\)

Answers (1)

2021-02-04
\(\lim_{n\rightarrow\infty}\frac{1}{3+ni}\)
Rationalize \(\frac{1}{3+ni}\) as
\(\frac{1}{3+ni}=\frac{1}{3+ni}\cdot\frac{3-ni}{3-ni}\)
\(=\frac{3-ni}{3^2-n^2i^2}\)
Substitute \(i^2=-1\) and solve further
\(\frac{1}{3+ni}=\frac{3-ni}{3^2+n^2}\)
\(=\frac{\frac{3-ni}{n^2}}{\frac{3^2+n^2}{n^2}}\)
\(=\frac{\frac{3}{n^2}-\frac{1}{ni}}{\frac{3^2}{n^2}+1}\)
Substitute
\(\frac{1}{3+ni}=\frac{\frac{3}{n^2}-\frac{1}{n}i}{\frac{3^2}{n^2}+1}\)
Hence
\(\lim_{n\rightarrow\infty}\frac{1}{3+ni}=\lim_{n\rightarrow\infty}\frac{\frac{3}{n^2}-\frac{1}{ni}}{\frac{3^2}{n^2}+1}\)
\(=\frac{(0-0i)}{(0+1)}\)
\(=0\)
0

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