"Correct Answer - Option 2 : e-2
The correct answer is option 2
Concept:
\(\displaystyle\lim_{n → a} \left(f(n) \right)^{g(n)}\) =\((\displaystyle\lim_{n → a} f(n) )^{\displaystyle\lim_{n → a}{g(n)}}\)
if f(n) → 1
And \(g(n) \rightarrow \infty\)
1∞ is one of the indeterminant forms.
Then convert it into exponential form
\(\displaystyle\lim_{n → a} \left(f(n) \right)^{g(n)}\) = \(e^{\displaystyle\lim_{n → a} \left(f(n)-1 \right)\times{g(n)}}\)
Calculation:
\(\displaystyle\lim_{n \rightarrow ∞} \left(1 - \dfrac{1}{n} \right)^{2n}\)
put \(n= \infty \)
⇒\( \left(1 - \dfrac{1}{\infty} \right)^{2\times\infty}\)
⇒ \( \left(1 - 0 \right)^{\infty}\)
⇒ \( \left(1 \right)^{\infty}\)
1∞ is one of the indeterminant forms.
Now, we convert it into exponential form
⇒ \(e^{\displaystyle\lim_{n → \infty} \left(1-\frac{1}{n}-1 \right)\times{2n}}\)
⇒ \(e^{\displaystyle\lim_{n → \infty} \left(-\frac{1}{n} \right)\times{2n}}\)
⇒ e-2
So, the correct answer is e-2limxeln(1+1xx→∞(1+1limx→∞eln(1+1x)x
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