Question

Find the limit as x approaches infinity of `y=arctan(e^x)`?

Answer 1

To answer this question, you need to know that `lim_{x to +infty} e^x=+infty` and `lim_{x to +infty} arctan x=pi/2` from the stuy of `e^x` (see Exponential functions ) and of `arctan x` (see inverse cosine and inverse tangent ).

So, as `x to infty`, `e^x to infty` so that, letting `t=e^x` we have
`lim_{x to infty} arctan(e^x)=lim_{t to infty} arctan(t)=\pi/2`.