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Geekchick
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I just wanted to know why the limit of arctanx as x approaches infinity is [tex]\frac{\pi}{2}[/tex]. It doesn't make any sense to me.
Geekchick said:Also, when determining if an improper integral diverges. It always diverges if the limit is infinity? my text doesn't say it just gives two examples where this is the case so I wanted to check before I generalized.
thanks!
Geekchick said:I just wanted to know why the limit of arctanx as x approaches infinity is [tex]\frac{\pi}{2}[/tex]. It doesn't make any sense to me.
arildno said:Since the mapping from angles to tangent values is bijective,...
The limit of arctanx as x approaches infinity is equal to
The limit of arctanx can be explained intuitively by considering the graph of the arctan function. As x approaches infinity, the graph of arctanx approaches a horizontal asymptote at This means that the values of arctanx get closer and closer to as x gets larger.
As x approaches infinity, the arctan function essentially "flattens out" and becomes a horizontal line at This is because as x gets larger, the ratio of the opposite side to the adjacent side of a right triangle becomes very small, essentially approaching 0. This means that the angle formed by this ratio approaches and thus, the value of arctanx approaches as x gets larger.
Yes, the limit of arctanx as x approaches infinity can be evaluated using L'Hopital's rule. By taking the derivative of both the numerator and denominator and applying the rule repeatedly, the limit can be simplified to
Yes, the limit of arctanx has applications in fields such as engineering, physics, and computer science. For example, it is used in calculating the phase difference between two signals in electrical engineering and in finding the angle of elevation in navigation and surveying. It is also used in the development of algorithms and computer graphics to determine the orientation of 3D objects.