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\(\lim _{x \rightarrow 0}(1+3 x)^{\frac{2}{x}}=\)
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Verified Answer
The correct answer is:
\(e^6\)
\(\begin{aligned}
\operatorname{Lim}_{x \rightarrow 0}(1 & +3 x) \frac{2}{x}=\operatorname{Lim}_{x \rightarrow 0}\left((1+3 x)^{\frac{1}{3 x}}\right)^6 \\
=e^6 \quad & \left\{\because \operatorname{Lim}_{x \rightarrow 0}(1+a x)^{\frac{1}{a x}}=e,(a \neq 0)\right\}
\end{aligned}\)
\operatorname{Lim}_{x \rightarrow 0}(1 & +3 x) \frac{2}{x}=\operatorname{Lim}_{x \rightarrow 0}\left((1+3 x)^{\frac{1}{3 x}}\right)^6 \\
=e^6 \quad & \left\{\because \operatorname{Lim}_{x \rightarrow 0}(1+a x)^{\frac{1}{a x}}=e,(a \neq 0)\right\}
\end{aligned}\)
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