Given,

\(\lim_{x\rightarrow0}\frac{\tan5x}{x}\)

On simplification, we get

\(\lim_{x\rightarrow0}\frac{\tan5x}{x}=\lim_{x\rightarrow0}\frac{5\tan5x}{5x}\)

\(=5\cdot\lim_{x\rightarrow0}\frac{\tan5x}{5x}\)

\(=5\cdot\lim_{5x\rightarrow0}\frac{\tan5x}{5x}\ \ [\because As\ x\rightarrow0,5x\rightarrow0]\)

\(=5\cdot(1)\ \ [\because\lim_{\theta\rightarrow0}\frac{\tan\theta}{\theta}=1]\)

\(=5\)

\(\lim_{x\rightarrow0}\frac{\tan5x}{x}\)

On simplification, we get

\(\lim_{x\rightarrow0}\frac{\tan5x}{x}=\lim_{x\rightarrow0}\frac{5\tan5x}{5x}\)

\(=5\cdot\lim_{x\rightarrow0}\frac{\tan5x}{5x}\)

\(=5\cdot\lim_{5x\rightarrow0}\frac{\tan5x}{5x}\ \ [\because As\ x\rightarrow0,5x\rightarrow0]\)

\(=5\cdot(1)\ \ [\because\lim_{\theta\rightarrow0}\frac{\tan\theta}{\theta}=1]\)

\(=5\)