Evaluate each limit and justify your answer. \lim_{x\to1}(\frac{x+5}{x+2}

babeeb0oL 2021-10-30 Answered
Evaluate each limit and justify your answer.
\(\displaystyle\lim_{{{x}\to{1}}}{\left({\frac{{{x}+{5}}}{{{x}+{2}}}}\right)}^{{4}}\)

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Expert Answer

Brittany Patton
Answered 2021-10-31 Author has 27655 answers
The given limit is:
\(\displaystyle\lim_{{{x}\to{1}}}{\left({\frac{{{x}+{5}}}{{{x}+{2}}}}\right)}^{{4}}\)
When substituting the direct limit, this limit will not undefined.
Therefore, to evaluate this take the limit directly.
That is,
\(\displaystyle\lim_{{{x}\to{1}}}{\left({\frac{{{x}+{5}}}{{{x}+{2}}}}\right)}^{{4}}={\left({\frac{{{1}+{5}}}{{{1}+{2}}}}\right)}^{{4}}\)
\(\displaystyle\lim_{{{x}\to{1}}}{\left({\frac{{{x}+{5}}}{{{x}+{2}}}}\right)}^{{4}}={\left({\frac{{{6}}}{{{3}}}}\right)}^{{4}}\)
\(\displaystyle\lim_{{{x}\to{1}}}{\left({\frac{{{x}+{5}}}{{{x}+{2}}}}\right)}^{{4}}={\left({2}\right)}^{{4}}\)
\(\displaystyle\lim_{{{x}\to{1}}}{\left({\frac{{{x}+{5}}}{{{x}+{2}}}}\right)}^{{4}}={16}\)
Answer: \(\displaystyle\lim_{{{x}\to{1}}}{\left({\frac{{{x}+{5}}}{{{x}+{2}}}}\right)}^{{4}}={16}\)
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