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 α = arctan(x) which gives tan α = x = x / 1 and use the definition of tan x in a right triangle to visualise x = tan α .

Right Triangle
Fig. 1 - α = arctan(x) in a Right Triangle

Geometrical Approach
As x increases indefinitely, geometrically α approaches π/2 and hence we write the limit
 Limit of Alpha

Algebraic Approach
We can also use sin (α) given by
 Sine of Alpha
Calculate the limit
 Calculate Limit
\( \) \( \)\( \)\( \) 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 inverse function \( y = \arctan (x) \) 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 \[ \lim_{x \to -\infty} \arctan(x) = -\dfrac{\pi}{2} \]

Graph of tan(x) and arctan(x)
Fig. 2 - Graph of \( \tan(x) \) and \( \arctan(x) \)

More References and Links

  1. Tangent Function tan x
  2. One to One Function
  3. Inverse Function
  4. Introduction to Limits