$\frac{ \lim_{x \rightarrow 0} \left(x \times \csc\left({2x}\right)\right) }{ \lim_{x \rightarrow 0} \left(\cos\left({5x}\right)\right) }$
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$\begin{align*}&\frac{ 1 }{ 2 } \\&\begin{array} { l }0.5,& {2}^{-1}\end{array}\end{align*}$