The limit of \(\arctan(x)\) as \(x\) approaches infinity is examined using two different approaches. The first one is based on the right triangle, and the second is based on the inverse function \(\arctan(x)\) definition.
\(\arctan(x)\) in a Right Triangle Approach
Let \(\alpha = \arctan(x)\) , which gives \(\tan \alpha = x = \dfrac{x}{1}\) and use the definition of tan in a right triangle to visualize \(x = \tan \alpha\) .
Fig. 1 - \(\alpha = \arctan(x)\) in a Right Triangle
Geometrical Approach
As \(x\) increases indefinitely, geometrically \(\alpha\) approaches \(\dfrac{\pi}{2}\) and hence we write the limit:
\[
\lim_{x \to \infty} \alpha = \dfrac{\pi}{2}
\]
Algebraic Approach
We can also use \(\sin(\alpha)\) given by:
\[
\sin(\alpha) = \dfrac{x}{\sqrt{1+x^2}}
\]
Calculate the limit:
\[
\lim_{x \to \infty} \sin(\alpha) = \lim_{x \to \infty} \dfrac{x}{\sqrt{1+x^2}} = 1
\]
which gives:
\[
\lim_{x \to \infty} \sin(\alpha) = \sin\dfrac{\pi}{2}
\]
Using the limit of composite functions:
\[
\lim_{x \to \infty} \sin(\alpha) = \sin\Big(\lim_{x \to \infty} \alpha\Big)
\]
Since \(\dfrac{\pi}{2}\) is a constant, we can write:
\[
\dfrac{\pi}{2} = \lim_{x \to \infty} \dfrac{\pi}{2}
\]
so that:
\[
\lim_{x \to \infty} \sin(\alpha) = \sin\Big(\lim_{x \to \infty} \alpha\Big) = \sin\Big(\lim_{x \to \infty} \dfrac{\pi}{2}\Big)
\]
Finally, using \(\alpha = \arctan(x)\):
\[
\lim_{x \to \infty} \alpha = \lim_{x \to \infty} \arctan(x) = \dfrac{\pi}{2}
\]
Tan(x) and Arctan(x) Graphs Approach
Below is shown the graph of the function \(y = \tan x\) (in blue) on the interval \(\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)\), with \(x = \dfrac{\pi}{2}\) and \(x = -\dfrac{\pi}{2}\) being vertical asymptotes (dashed blue line). It is a one-to-one function and therefore has the inverse function \(y = \arctan(x)\) as shown below (in red).
The vertical asymptotes of \(y = \tan x\) become horizontal asymptotes (dashed red line) of the inverse function \(y = \arctan(x)\), which by definition may be written using limits as follows:
\[
\lim_{x \to \infty} \arctan(x) = \dfrac{\pi}{2}, \quad
\lim_{x \to -\infty} \arctan(x) = -\dfrac{\pi}{2}
\]
Fig. 2 - Graph of \( \tan(x) \) and \( \arctan(x) \)
