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

Find the following complex limits. $\underset{n\to \mathrm{\infty }}{lim}\frac{2n}{7n+ni}$
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Velsenw
$\underset{n\to \mathrm{\infty }}{lim}\frac{2n}{7n+ni}$
$\frac{2n}{7n+ni}=\frac{\frac{2n}{n}}{\frac{7n+ni}{n}}$
$=\frac{2}{7+i}$
Rationalize $\frac{2}{7+i}$ as
$\frac{2}{7+i}=\frac{2}{7+i}\cdot \frac{7-i}{7-i}$
$=\frac{2\left(7-i\right)}{49-{i}^{2}}$
$=\frac{2\left(7-i\right)}{49+1}$
$=\frac{2\left(7-i\right)}{50}$
$=\frac{7-i}{25}$
Substitute $\frac{2n}{7n+ni}=\frac{7-i}{25}$ in $\underset{n\to \mathrm{\infty }}{lim}\frac{2n}{7n+ni}$
$\underset{n\to \mathrm{\infty }}{lim}\frac{2n}{7n+ni}=\underset{n\to \mathrm{\infty }}{lim}\frac{7-i}{25}$
$=\frac{7-i}{25}$
$=\frac{7}{25}-\frac{1}{25i}$