A tangent line is NEVER vertical to a function. It is vertical to the normal to the function - which is as far from vertical as you can get!
The graph of a function, f(x) can have a tangent at a point. Let's call the point (x0,f(x0)). If f'(x) goes to positive infinity or f'(x) goes to negative infinity as x approaches x0 then f(x) has a vertical tangent at that point.