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. [br]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?
The functions[br][img]data:image/png;base64,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[/img][br]have 3 points where the [math]lim\frac{f\left(x\right)}{g\left(x\right)}[/math] is undefined. [br][br]Please identify the three points explicitly and the limit at each of those points.
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Information: Visualizing L'Hopital's Rule