The value of \[\mathop {\lim }\limits_{{\text{x}} \to \infty } {\left( {1 + \frac{1}{{\text{x}}}} \right)^{\text{x}}}\] is A. \[l{\text{n}}2\] B. 1.0 C. e D. \[\infty \]

” option2=”1″ option3=”e” option4=”\[\infty \]” correct=”option1″]

The correct answer is $\boxed{\text{C}}$.

Let’s take a look at each option:

• A. $\ln 2$

$\ln 2$ is the natural logarithm of 2, which is approximately equal to 0.6931471805599453. As $x$ approaches infinity, $\left(1+\frac{1}{x}\right)^x$ approaches $\ln 2$. However, it does not equal $\ln 2$.

• B. 1.0

1.0 is the value of $\left(1+\frac{1}{x}\right)^x$ when $x=1$. However, as $x$ approaches infinity, $\left(1+\frac{1}{x}\right)^x$ approaches $\ln 2$, which is greater than 1.0.

• C. $e$

$e$ is an irrational and transcendental constant approximately equal to 2.71828. It is the base of the natural logarithm and the exponential function. As $x$ approaches infinity, $\left(1+\frac{1}{x}\right)^x$ approaches $e$.

• D. $\infty$

$\infty$ is an indeterminate form that represents an infinite number. As $x$ approaches infinity, $\left(1+\frac{1}{x}\right)^x$ approaches $\infty$. However, it does not equal $\infty$.

Therefore, the correct answer is $\boxed{\text{C}}$.