# Limits of composite functions Evaluate each limit and justify your answer PS

$$\displaystyle\lim_{{{x}\to{4}}}{\tan{{\frac{{{t}-{4}}}{{\sqrt{{{t}}}-{2}}}}}}$$

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2abehn
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 $$\displaystyle{\tan{{4}}}$$