Evaluate limx→∞tan−1(x) using that y=tan−1(x) exactly when x=tan(y) . (Hint: Both tan and tan−1 are continuous!)

to \:evaluate \:the \:limit \:of\: \tan^{-1}\left(x\right)\: as \:x approaches \:infinity we \:can \:use\: the\: fact\: that\: y = \tan^{-1}\left(x\right) \:implies\: x \:= \:tan(y). As x approaches infinity, y will also approach a certain value. We can determine this value by considering the behavior of the tangent function as the angle increases without bound. As y approaches 90 degrees from below, the value of tan(y) approaches positive infinity. Therefore, as x approaches infinity, \:y \:approaches\: \frac{π}{2}. Using the relationship x = tan(y), we can say that as x approaches infinity, y \:approaches\: \frac{π}{2}. Therefore, \:the \:limit\: of \:\tan^{-1}\left(x\right)\: as\: x\: approaches\: infinity\: is\: \frac{π}{2} or 90^\circ.