Visualizing l'Hopital's Rule
Move the slider or watch the animation, and observe that the ratio of the slopes of the tangent lines approaches the ratio of the values of the functions at the x-intercepts.
As the tangent lines approach each x-intercept, can you see how the ratio of their slopes ought to be a close approximation of the ratio of their values as L'Hopital's Rule would indicate?
Type new functions if you like, but if you do, the tangent lines may not keep up with you.
To fix this, (1) move the slider, then (2) click the undo button.
Then the tangent lines should catch up with you.