Several studies on student's understanding of the concept of the tangent line agree that the difficulties students have when dealing with lines tangent to curves at inflection points or tangent lines that have more than one point in common with the curve are due to a misconception of tangency that is limited to the geometric relationship between line and circle. Here, we report difficulties Mexican high school students had when working on tangent lines in situations like those mentioned before. For our research, we designed Activities mostly carried out with dynamic geometry. Using the zoom-in provided by the software helped strengthen the student's mental image of the local geometric relationship between a curve and a line tangent. Several of the curves we designed for the investigation surprised the students when they zoomed in on the point of tangency. From the point of view of the tangency relation that is intuitively obtained from tangents to the circle or curves at inflection points, in several curves that we presented, the tangent lines neither touch nor cross the curves at the points of tangency. These are cases of curves and tangents that would be worth analyzing in Differential Calculus courses.