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 \( x = \tan \alpha \).
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 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
\[ \lim_{x \to -\infty} \arctan(x) = -\dfrac{\pi}{2} \]
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