# Find the limit. lim_(xrightarrow 0)((x csc 2x))/(cos 5x)

Find the limt. $\underset{x\to 0}{lim}﻿\frac{\left(x\mathrm{csc}2x\right)}{\mathrm{cos}5x}$
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Jaelyn Rosario
$\underset{x\to 0}{lim}\frac{\left(x\mathrm{csc}2x\right)}{\mathrm{cos}5x}=\underset{x\to 0}{lim}\frac{x}{\mathrm{sin}2x\left(\mathrm{cos}5x\right)}$
$=\underset{x\to 0}{lim}\frac{1}{2\mathrm{cos}2x\ast \left(\mathrm{cos}5x\right)-5\mathrm{sin}2x\mathrm{sin}5x}=\frac{1}{2\mathrm{cos}\left(2\ast 0\right)\mathrm{cos}\left(5\ast 0\right)-5\ast \mathrm{sin}\left(2\ast 0\right)\ast \mathrm{sin}\left(5\ast 0\right)}$
$=\frac{1}{2\ast 1\ast 1-5\ast 0\ast 0}=\frac{1}{2}$
###### Not exactly what you’re looking for?
Lacey Rojas
$li{m}_{x\to 0}\left(x\mathrm{csc}\left(2x\right)\right)/\left(\mathrm{cos}\left(5x\right)=\underset{x\to 0}{lim}\left(1/2\right)\left(2x/\mathrm{sin}\left(2x\right)\right)/\mathrm{cos}\left(5x\right)$
Now
so $\underset{x\to 0}{lim}\left(x\mathrm{csc}\left(2x\right)\right)/\left(\mathrm{cos}\left(5x\right)=1/2$