Choose one of the following topics:

- The precise definition of a limit
- A list of basic limit laws
- The statement of l'Hôpital's Rule
- A quick note on sequences

Or, choose one of the following "forms" of limit, to which l'Hôpital's Rule can be applied:

- lim
*f*(*x*) /*g*(*x*), where both*f*(*x*) and*g*(*x*) approach zero - lim
*f*(*x*) /*g*(*x*), where both*f*(*x*) and*g*(*x*) approach infinity - lim
*f*(*x*) **g*(*x*), where*f*(*x*) approaches infinity and*g*(*x*) approaches zero - lim
*f*(*x*) -*g*(*x*), where both*f*(*x*) and*g*(*x*) approach infinity - lim
*f*(*x*)^*g*(*x*), where both*f*(*x*) and*g*(*x*) approach zero - lim
*f*(*x*)^*g*(*x*), where*f*(*x*) approaches infinity and*g*(*x*) approaches zero - lim
*f*(*x*)^*g*(*x*), where*f*(*x*) approaches one and*g*(*x*) approaches infinity

does not have a value when *x*=2. However, if you inspect the function
values for values of *x* near 2, you will discover a trend:

f(1.5)
| = | 2.5 | f(2.5)
| = | 3.5 |

f(1.7)
| = | 2.7 | f(2.3)
| = | 3.3 |

f(1.8)
| = | 2.8 | f(2.2)
| = | 3.2 |

f(1.9)
| = | 2.9 | f(2.1)
| = | 3.1 |

f(1.99)
| = | 2.99 | f(2.01)
| = | 3.01 |

f(1.999)
| = | 2.999 | f(2.001)
| = | 3.001 |

As you can see, as *x* gets closer
to 2, the function
value *f*(*x*) gets closer to 3. We say that the limit of
*f*(*x*) as *x* approaches 2 is 3, or symbolically,

.

*l'Hôpital's Rule* is a mathematical tool which is often quite
useful for finding limits. As its most straightforward use, it is helpful
for certain fractions which, otherwise, would require much more work to
find their limits. However, using algebraic manipulation, there are many
other forms of limits for which it can be helpful.

Choose one of the following topics:

- The precise definition of a limit
- A list of basic limit laws
- The statement of l'Hôpital's Rule
- A quick note on sequences

Or, choose one of the following "forms" of limit, to which l'Hôpital's Rule can be applied:

- lim
*f*(*x*) /*g*(*x*), where both*f*(*x*) and*g*(*x*) approach zero - lim
*f*(*x*) /*g*(*x*), where both*f*(*x*) and*g*(*x*) approach infinity - lim
*f*(*x*) **g*(*x*), where*f*(*x*) approaches infinity and*g*(*x*) approaches zero - lim
*f*(*x*) -*g*(*x*), where both*f*(*x*) and*g*(*x*) approach infinity - lim
*f*(*x*)^*g*(*x*), where both*f*(*x*) and*g*(*x*) approach zero - lim
*f*(*x*)^*g*(*x*), where*f*(*x*) approaches infinity and*g*(*x*) approaches zero - lim
*f*(*x*)^*g*(*x*), where*f*(*x*) approaches one and*g*(*x*) approaches infinity

Copyright © 1996 Department of Mathematics, Oregon State University

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