# Limits and l'Hôpital's Rule

Choose one of the following topics:

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

Limits are a mathematically precise way of representing how a function acts when its independent variable is close to a certain number. For instance, the function

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:

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