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 $\boxed{\text{C}}$.