When analyzing tan(x), I have found it simple to translate tan(x) to sin(x)/cos(x). If you do not understand why tan(x)=sin(x)/cos(x), read the next paragraph:
On the unit circle, all y-coordinate values represent sin and all x-coordinate values represent cos.Tan is the slope of any given coordinate of sin and cos on the unit circle. We know that slope equals y/x so tan(x)=sin(x)/cos(x).
So to solve these problems without a graphing calculator, substitute the given x-values into sin(x)/cos(x) to get the answer.
Step 1: f(π/2)=tan(π/2)
Step 2. tan(π/2)= sin(π/2)/cos(π/2)
Step 3. sin(π/2)= 1
Step 4: cos(π/2)= 0
Step 5: sin(π/2)/cos(π/2)= 1/0 = Does Not Exist
Hint: Tangent has a vertical asymptote at aπ/2 where a is any integer excluding zero and any multiple of 2π