Find the following complex limits. lim_{nrightarrowinfty}frac{6n}{2n-n^2i}

Find the following complex limits. lim_{nrightarrowinfty}frac{6n}{2n-n^2i}

Question
Limits and continuity
asked 2021-02-02
Find the following complex limits. \(\lim_{n\rightarrow\infty}\frac{6n}{2n-n^2i}\)

Answers (1)

2021-02-03
\(\lim_{n\rightarrow\infty}\frac{6n}{2n-n^2i}=\lim_{n\rightarrow\infty}\frac{6}{2n-ni}\)
Rationalize \(\frac{6}{2-ni}\) as
\(\frac{6}{2-ni}=\frac{6}{2-ni}\cdot\frac{2+ni}{2+ni}\)
\(=\frac{12+6ni}{4-n^2i^2}\)
\(=\frac{12+6ni}{4+n^2}\)
\(=\frac{\frac{12+6ni}{n^2}}{\frac{4+n^2}{n^2}}\)
\(=\frac{\frac{12}{n^2}+\frac{6}{ni}}{\frac{4^2}{n^2}+1}\)
Substitute \(\frac{6}{2-ni}=\frac{12}{n^2}+\frac{\frac{6}{ni}}{\frac{(4^2)}{n^2+1}}\)
\(=\frac{0+0i}{(0+1)}\)
\(=0\)
0

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