Evaluate: \lim_{x arrow 0} ((x * csc(2x))/(cos(5x)))

Expression: $\lim_{x \rightarrow 0} \left(\frac{ x \times \csc\left({2x}\right) }{ \cos\left({5x}\right) }\right)$

Use $\lim_{x \rightarrow c} \left(\frac{ f\left( x \right) }{ g\left( x \right) }\right)=\frac{ \lim_{x \rightarrow c} \left(f\left( x \right)\right) }{ \lim_{x \rightarrow c} \left(g\left( x \right)\right) }$ to transform the expression

$\frac{ \lim_{x \rightarrow 0} \left(x \times \csc\left({2x}\right)\right) }{ \lim_{x \rightarrow 0} \left(\cos\left({5x}\right)\right) }$

Use $\csc\left({t}\right)=\frac{ 1 }{ \sin\left({t}\right) }$ to transform the expression

$\frac{ \lim_{x \rightarrow 0} \left(x \times \frac{ 1 }{ \sin\left({2x}\right) }\right) }{ \lim_{x \rightarrow 0} \left(\cos\left({5x}\right)\right) }$

Use $\lim_{x \rightarrow c} \left(\cos\left({f\left( x \right)}\right)\right)=\cos\left({\lim_{x \rightarrow c} \left(f\left( x \right)\right)}\right)$ to transform the expression

$\frac{ \lim_{x \rightarrow 0} \left(x \times \frac{ 1 }{ \sin\left({2x}\right) }\right) }{ \cos\left({\lim_{x \rightarrow 0} \left(5x\right)}\right) }$

Calculate the product

$\frac{ \lim_{x \rightarrow 0} \left(\frac{ x }{ \sin\left({2x}\right) }\right) }{ \cos\left({\lim_{x \rightarrow 0} \left(5x\right)}\right) }$

Use $\lim_{x \rightarrow c} \left(a \times f\left( x \right)\right)=a \times \lim_{x \rightarrow c} \left(f\left( x \right)\right)$ to transform the expression

$\frac{ \lim_{x \rightarrow 0} \left(\frac{ x }{ \sin\left({2x}\right) }\right) }{ \cos\left({5 \times \lim_{x \rightarrow 0} \left(x\right)}\right) }$

Since evaluating limits of the numerator and denominator would result in an indeterminate form, use the L'Hopital's rule

$\frac{ \lim_{x \rightarrow 0} \left(\frac{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( x \right) }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( \sin\left({2x}\right) \right) }\right) }{ \cos\left({5 \times \lim_{x \rightarrow 0} \left(x\right)}\right) }$

Evaluate the limit by substituting the value $x=0$ into the expression

$\frac{ \lim_{x \rightarrow 0} \left(\frac{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( x \right) }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( \sin\left({2x}\right) \right) }\right) }{ \cos\left({5 \times 0}\right) }$

Find the derivative

$\frac{ \lim_{x \rightarrow 0} \left(\frac{ 1 }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( \sin\left({2x}\right) \right) }\right) }{ \cos\left({5 \times 0}\right) }$

Find the derivative

$\frac{ \lim_{x \rightarrow 0} \left(\frac{ 1 }{ 2\cos\left({2x}\right) }\right) }{ \cos\left({5 \times 0}\right) }$

Evaluate the limit

$\frac{ \frac{ 1 }{ 2\cos\left({2 \times 0}\right) } }{ \cos\left({5 \times 0}\right) }$

Simplify the expression

$\begin{align*}&\frac{ 1 }{ 2 } \\&\begin{array} { l }0.5,& {2}^{-1}\end{array}\end{align*}$