Find the limit as x -> infinity of arctan(e^x)

[MarkSA]

Hello,

Find the limit as x -> infinity, of: arctan(e^x)

In this problem, i'm not entirely sure of the proper process to use to get the answer. If I just work with the inner part, I suppose I can say that as x approaches infinity, e^x approaches infinity. Then, would I just determine what arctan(x) does as x approaches infinity? Is there a way to determine this without a graphing calculator? I have the general idea what sin,cos,tan functions look like, but I have trouble picturing what the inverses would look like (reflected about the line y=x.) I know in this case the limit is pi/2, but i'm not sure how I would do other problems like this without a graphing calculator.

 

[tkhunny]

This is a definition problem.

y = tan(x) has asymptotes at the same places as cos(x) = 0, as in -pi/2 and pi/2, for example. There are lots more pieces of the tangent function, but this one is very important.

y = atan(x) - the inverse FUNCTION of the tangent function, uses only the one piece mentioned above, otherwise it isn't a function. The RANGE of the inverse tangent function is (-pi/2, pi/2). That's just the definition.

Whatever that argument is, be it x, e^x, ln(x), 2x^2, 1+x, if it is increasing without bound, the inverse tangent function will be approaching pi/2.

You would do other related problems by knowing the definitions and why they are so.