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

Bevan Mcdonald 2021-10-26 Answered
Limits of composite functions Evaluate each limit and justify your answer
\(\displaystyle\lim_{{{x}\to{4}}}{\tan{{\frac{{{t}-{4}}}{{\sqrt{{{t}}}-{2}}}}}}\)

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Expert Answer

2abehn
Answered 2021-10-27 Author has 13622 answers
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 \(\displaystyle{\tan{{4}}}\)
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