Limits and l'Hôpital's Rule

• Return to the Sequences and Series starting page

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.

• Return to the Sequences and Series starting page

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

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