We have to find the limits by using L 'Hospital's rule:

\(\lim_{x\rightarrow0}\frac{\sin mx}{\sin nx}\)

In L' Hospital's rule we differentiate numerator as well as denominator if they have the form \(\frac{0}{0}\ and\ \frac{\infty}{\infty}\)

After putting limits value we can say that this is \(\frac{0}{0}\) form therefore we can successfully apply the rule so differentiating numerator and denominator with respect to x,

\(\lim_{x\rightarrow0}\frac{\sin mx}{\sin nx}=\lim_{x\rightarrow0}\frac{\frac{d(\sin mx)}{dx}}{\frac{d(\sin nx)}{dx}}\)

We know that

\(\frac{d(\sin ax)}{dx}=\cos ax\frac{d(ax)}{dx}\)

\(=\cos ax(a\frac {dx}{dx})\)

\(=a\cos ax\)

Solving further using above formula,

\(\lim_{x\rightarrow0}\frac{\frac{d(\sin mx)}{dx}}{\frac{d(\sin nx)}{dx}}=\lim_{x\rightarrow0}\frac{\cos mx(\frac{d(mx)}{dx})}{\cos nx(\frac{d(nx)}{dx})}\)

\(=\lim_{x\rightarrow0}\frac{\cos mx(m\frac{dx}{dx})}{\cos nx(n\frac{dx}{dx})}\)

\(=\lim_{x\rightarrow0}\frac{\cos mx(m\times1)}{\cos nx(n\times1)}\)

\(=\frac{m\cos m\times0}{n\cos n\times0}\)

\(=\frac{m\cos0}{n\cos0}\)

\(=\frac{m\times1}{n\times1}\)

\(=\frac{m}{n}\)

Hence, value of limit is \(=\frac{m}{n}\)

\(\lim_{x\rightarrow0}\frac{\sin mx}{\sin nx}\)

In L' Hospital's rule we differentiate numerator as well as denominator if they have the form \(\frac{0}{0}\ and\ \frac{\infty}{\infty}\)

After putting limits value we can say that this is \(\frac{0}{0}\) form therefore we can successfully apply the rule so differentiating numerator and denominator with respect to x,

\(\lim_{x\rightarrow0}\frac{\sin mx}{\sin nx}=\lim_{x\rightarrow0}\frac{\frac{d(\sin mx)}{dx}}{\frac{d(\sin nx)}{dx}}\)

We know that

\(\frac{d(\sin ax)}{dx}=\cos ax\frac{d(ax)}{dx}\)

\(=\cos ax(a\frac {dx}{dx})\)

\(=a\cos ax\)

Solving further using above formula,

\(\lim_{x\rightarrow0}\frac{\frac{d(\sin mx)}{dx}}{\frac{d(\sin nx)}{dx}}=\lim_{x\rightarrow0}\frac{\cos mx(\frac{d(mx)}{dx})}{\cos nx(\frac{d(nx)}{dx})}\)

\(=\lim_{x\rightarrow0}\frac{\cos mx(m\frac{dx}{dx})}{\cos nx(n\frac{dx}{dx})}\)

\(=\lim_{x\rightarrow0}\frac{\cos mx(m\times1)}{\cos nx(n\times1)}\)

\(=\frac{m\cos m\times0}{n\cos n\times0}\)

\(=\frac{m\cos0}{n\cos0}\)

\(=\frac{m\times1}{n\times1}\)

\(=\frac{m}{n}\)

Hence, value of limit is \(=\frac{m}{n}\)