# What is the value of $\displaystyle\lim_{n \rightarrow \infty} \left(1 - \dfrac{1}{n} \right)^{2n}$ ?

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What is the value of $\displaystyle\lim_{n \rightarrow \infty} \left(1 - \dfrac{1}{n} \right)^{2n}$ ?

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"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

"