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\(a_{n} =(1+\frac{5}{3n})^{7n}\)
\(b_{n} =(1-\frac{6}{n^{2}})^{ n^{2}} \\
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c_{n}=(\frac{n+3}{n-3} )^{2n+3}\\
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d_{n}=(1+\frac{115}{4n+3})^{n}\\
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e_{n}=(\frac{2+3n}{1+3n})^{4n-1}\\
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f_{n}=(\frac{n^{2}-1}{n^{2}+2})^{ n^{2}}\\
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g_{n}=(\frac{n^{2}-1}{2n^{2}+2})^{ n^{2}}\\
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h_{n}=( \frac{2n^{2}-1}{n^{2}+2} )^{n^{2}}\)

\(\Lim_{n\to \infty }a_n=\Lim_{n\to\infty}((1+ \frac{ \frac{5}{3} }{n})^n)^7=(e^{ \frac{5}{3} })^7=e^{ \frac{35}{3} }\)
\(\Lim_{n\to \infty }b_n= \Lim_{n\to \infty }(1+ \frac{-6}{n^2})^{n^2}=e^{-6}= \frac{1}{e^6}\)
\(\Lim_{n\to \infty }c_n= \Lim_{n\to \infty }[(1+ \frac{6}{n-3})^{n-3}]^{ \frac{2n+3}{n-3} }=[e^6]^2=e^{12}\)
\(\Lim_{n\to\infty }d_n= \Lim_{n\to \infty } [(1+ \frac{115}{4n+3})^{4n+3}]^{ \frac{n}{4n+3} }=[e^{115}]^{ \frac{1}{4} }=e^{ \frac{115}{4} }\)
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