Evaluate the limit: _x infinity (1 − 2/x)^x

Question

ScanSolve AI Tutor · Accepted Answer

Let's find the limit of the given function.

The problem asks to evaluate the limit:

\lim_{x \to \infty} \left(1 - \frac{2}{x}\right)^x

Recognize the form of the limit.
This limit is in the indeterminate form $1^\infty$ as $x \to \infty$ , since $\left(1 - \frac{2}{x}\right) \to (1 - 0) = 1$ . This form is related to the definition of the mathematical constant $e$ .
Apply the standard limit formula for $e$ .
A common limit definition for $e$ is:
$\lim_{x \to \infty} \left(1 + \frac{k}{x}\right)^x = e^k$
where $k$ is a constant.
Compare the given limit with the standard formula.
In our problem, we have $\left(1 - \frac{2}{x}\right)^x$ . We can rewrite this as $\left(1 + \frac{-2}{x}\right)^x$ .
By comparing this to the standard formula, we can identify $k = -2$ .
Substitute the value of $k$ into the formula.
Using $k = -2$ :
$\lim_{x \to \infty} \left(1 + \frac{-2}{x}\right)^x = e^{-2}$
Express the result with a positive exponent.
$e^{-2} = \frac{1}{e^2}$
The final answer is:
$\frac{1}{e^2}$

Evaluate the limit: _x infinity (1 - (2)/(x))^x