To evaluate the limit $\lim_{{x \to \infty}} \frac{\arctan(x)}{x}$, let's follow these steps:

Step 1: Behavior of $\arctan(x)$ as $x \to \infty$

The arctangent function $\arctan(x)$ approaches $\frac{\pi}{2}$ as $x \to \infty$. So:

$\lim_{{x \to \infty}} \arctan(x) = \frac{\pi}{2}$

Step 2: Apply the limit to the given expression

Now, let's consider the ratio $\frac{\arctan(x)}{x}$. As $x$ grows very large:

• The numerator, $\arctan(x)$, approaches $\frac{\pi}{2}$, which is a constant.
• The denominator, $x$, grows without bound.

This means we are essentially dividing a constant by an infinitely large number:

[ \lim_{{x \to \infty}} \frac{\arct