Given limit is:

\(\displaystyle{L}=\lim_{{{x}\to{4}}}{\tan{{\frac{{{t}-{4}}}{{\sqrt{{{t}}}-{2}}}}}}\)

Then we get,

\(\displaystyle{L}=\lim_{{{t}\to{4}}}{\left[{\tan{{\frac{{{\left(\sqrt{{{t}}}+{2}\right)}{\left(\sqrt{{{t}}}-{2}\right)}}}{{\sqrt{{{t}}}-{2}}}}}}\right]}\)

\(\displaystyle{L}=\lim_{{{t}\to{4}}}{\left[{\tan{{\left(\sqrt{{{t}}}+{2}\right)}}}\right]}\)

\(\displaystyle{L}={\tan{{\left(\sqrt{{{4}}}+{2}\right)}}}\)

\(\displaystyle{L}={\tan{{\left({2}+{2}\right)}}}\)

\(\displaystyle{L}={\tan{{4}}}\)

Hence the value of this limit is \(\displaystyle{\tan{{4}}}\)

\(\displaystyle{L}=\lim_{{{x}\to{4}}}{\tan{{\frac{{{t}-{4}}}{{\sqrt{{{t}}}-{2}}}}}}\)

Then we get,

\(\displaystyle{L}=\lim_{{{t}\to{4}}}{\left[{\tan{{\frac{{{\left(\sqrt{{{t}}}+{2}\right)}{\left(\sqrt{{{t}}}-{2}\right)}}}{{\sqrt{{{t}}}-{2}}}}}}\right]}\)

\(\displaystyle{L}=\lim_{{{t}\to{4}}}{\left[{\tan{{\left(\sqrt{{{t}}}+{2}\right)}}}\right]}\)

\(\displaystyle{L}={\tan{{\left(\sqrt{{{4}}}+{2}\right)}}}\)

\(\displaystyle{L}={\tan{{\left({2}+{2}\right)}}}\)

\(\displaystyle{L}={\tan{{4}}}\)

Hence the value of this limit is \(\displaystyle{\tan{{4}}}\)