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

Question
Limits and continuity
Find the following complex limits. $$\lim_{n\rightarrow\infty}\frac{1}{3+ni}$$

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
$$\lim_{n\rightarrow\infty}\frac{1}{3+ni}=\lim_{n\rightarrow\infty}\frac{\frac{3}{n^2}-\frac{1}{ni}}{\frac{3^2}{n^2}+1}$$
$$=\frac{(0-0i)}{(0+1)}$$
$$=0$$

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