Suppose that we want to find the value of

when *f* (*a*) = *g* (*a*) = 0.
One method is to use L'Hopital's Rule, which says:

## Example 1

## Example 2

## Example 3

NOTE: L'Hopital's Rule also works if *f* (*a*) = *g* (*a*) = ∞, and when* *.
## Other 'Applications of Differentiation' topics

- If
*f*(x) and*g*(*x*) are differentiable functions and *f*(*a*) =*g*(*a*) = 0, then- if the limit on the right exists.

See About the calculus applets for operating instructions. |

In the graph on the left, the applet shows a graph of
This is not defined for *x* = 0, but it clearly seems to have a limit there. If we let *f* (*x*) = *e*^{2x} - 1 and *g* (*x*) = *x*,*x* = 0, the ratio of the derivatives is defined, hence we can evaluate the limit just by substituting *x* = 0.

Why does this work? The graph on the right shows *f* (*x*) and *g* (*x*). If you click the zoom in button several times, local linearity makes the curves look like straight lines. Hence the functions can be approximated by their tangent lines at *x* = 0 (i.e., *f* (*x*) ≈ 2*x* and *g*(*x*) ≈ *x*) , and the ratio can be approximated by the ratio of these tangent lines. In the limit, this equals the ratio of the slopes of the tangent lines, which is just the ratio of the derivatives of the numerator and denominator functions

Select the second example from the drop down menu. This shows another example, Again, if you click on Zoom In a few times, the graphs look like straight lines and the ratio can be evaluated just by using the ratio of the values of the derivatives.

Select the third example, showing In this case, the limit on the right does not exist, so L'Hopital's Rule cannot be used in this case.

- Curve Analysis: Basics
- Curve Analysis: Special Cases
- Curve Analysis: Global Extrema
- Optimization: Maximize Volume
- Extreme Value Theorem
- Related Rates
- L'Hopital's Rule
- Parametric Derivatives
- Polar Derivatives
- Motion on a Line
- Motion in the Plane

(C) 2011 Copyright Math Open Reference.

All rights reserved

All rights reserved