Question

Find the limit x->pi/2^+ tan(x)

I know the answer is -∞, but I don't understand why. When I simplify to sin(x)/cos(x), I get 1/0^+.

Answer 1

tan=sin(x)/cos(x) is a good start.
How about putting
lim x&rarrow;pi+
as
x&rarrow; as (π/2+δ) as δ-> 0+.
tan(x)
=sin(x)/cos(x)
=sin(π/2+δ)/cos(π/2+δ)
=(sin(π/2)cos(δ)+cos(π/2)sin(δ))/ (cos(π/2)cos(δ)-sin(π/2)sin(δ))
setting sin(π/2)=1 and cos(π/2)=0,
=(cos(δ)+0*sin(δ))/ (0*cos(δ)-sin(δ))
=cos(δ)/(-sin(δ)
and setting cos(δ)=1, sin(δ)=0 as δ->+0
=1/(-0)
=-∞

Answer 2

π/2 is 90º

on the unit circle ... tan = y/x

at 90º, x = 0

dividing by zero gives the infinite limit

depending on the direction of approach to π/2, the sign of the limit can be +/-