# Limit of Arctan(x) as x Approaches Infinity

      

The lmit 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 of 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 x in a right triangle to visualise $$x = \tan \alpha$$. 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 $\lim_{x \to \infty} \alpha = \lim_{x \to \infty} \arctan(x) = \dfrac{\pi}{2}$

Algebraic Approach
We can also use $$\sin \alpha$$ given by $\sin \alpha = \dfrac{\text{opposite side}}{\text{hypotenuse}}= \dfrac{x}{\sqrt{1+x^2}}$ Calculate the limit
\begin{align} &\lim_{x \to \infty} \sin \alpha = \lim_{x \to \infty} \dfrac{x}{\sqrt{1+x^2}} \15pt] &= \lim_{x \to \infty} \dfrac{x}{\sqrt{x^2}} \\[15pt] &= \lim_{x \to \infty} \dfrac{x}{|x|} \\[15pt] &= 1 \\[15pt] &= \sin \dfrac{\pi}{2} \end{align} which gives \[ \lim_{x \to \infty} \sin \alpha = \sin \dfrac{\pi}{2} Using the limit of composite functions, we write $$\lim_{x \to \infty} \sin \alpha = \sin (\lim_{x \to \infty} \alpha)$$ and the fact that
$$\dfrac{\pi}{2}$$ is a constant hence $$\dfrac{\pi}{2} = \lim_{x \to \infty} \dfrac{\pi}{2}$$, we write $\lim_{x \to \infty} \sin \alpha = \sin (\lim_{x \to \infty} \; \alpha ) = \sin \lim_{x \to \infty} \dfrac{\pi}{2}$ Using $$\alpha = \arctan(x)$$ and the above, we write $\lim_{x \to \infty} \; \alpha = \lim_{x \to \infty} \; \arctan(x) = \dfrac{\pi}{2}$

## Tan (x) and Arctan(x) Graphs Approach

Below is shown the graph of the function function $$y = \tan x$$ (in blue) on the interval $$(-\dfrac{\pi}{2} , \dfrac{\pi}{2})$$, with $$x = \dfrac{\pi}{2}$$ and $$x = - \dfrac{\pi}{2}$$ being vertical asymptotes (broken line blue). It is a one to one function function and therefore has the $$y = \arctan (x)$$ as shown below (in red).
The vertical asymptotes of $$y = \tan x$$ becomes horizontal asymptotes (broken line red) of the inverse function $$y = \arctan (x)$$ which by definition may be written using limits as follows:
$\lim_{x \to \infty} \arctan(x) = \dfrac{\pi}{2}$ and $\lim_{x \to -\infty} \arctan(x) = -\dfrac{\pi}{2}$ Fig. 2 - Graph of $$\tan(x)$$ and $$\arctan(x)$$