Given information:

The expression limx→∞arctan(ex) .

Formula used:

Inverse tangent trigonometric function is expressed as tan−1(x)=y which is equivalent to x=tany where −π2<y<π2 .

Domain of inversetangent trigonometric function is (−∞,∞) and range is (−π2,π2) .

Calculation:

Consider the expression limx→∞arctan(ex) .

Recall that domain of inversetangent trigonometric function is (−∞,∞) and range is (−π2,π2) .

Substitute x as ∞ in the above expression and simplify,

  limx→∞arctan(ex)=arctan(e∞)=arctan(∞)

Recall that the inverse tangent trigonometric function is expressed as tan−1(x)=y which is equivalent to x=tany where −π2<y<π2 .

Recall that value of tangent function at an angle π2 is ∞ .

That is, tan(π2)=∞ .

Take tangent function to the right hand side,

  tan−1(∞)=π2 .

Therefore,

  limx→∞arctan(ex)=arctan(e∞)=arctan(∞)=π2

Thus, the value of trigonometric expression limx→∞arctan(ex) is π2 .