1. L’Hopital’s Rule Limits of the form correspond to undetermined forms of limits since the result depends upon the expression in consideration. L’Hopital’s rule tells how to deal with this type of limits using derivatives

2. Exercise Determine whether the following limits are of the form or neither

3. Limits of the form Evaluate the following limits

4. Using the Tangent Lines f(x) Tangent line to f(x) at x=1 g(x) Tangent line to g(x) at x=1

5. Using the Tangent Lines Tangent line to g(x) at x=1 f(x) Tangent line to f(x) at x=1 g(x)

6. First Version of L’Hopital’s Rule

7. The limit at x=1 is the same as the quotient of the derivatives at x=1

8. L’Hopital’s Rule

9. In a simpler language it says that to calculate the limit of the quotient of two functions at a point a, where the limit is of the form " 0”/" 0”, it is the same as the limit of the quotient of their derivatives at that point.

12. L'Hopital's rule also holds if we replace the limiting point a for ∞, -∞, a +, a -

13. Practice Calculate the following limits. In case you want to use L’Hopital’s rule verify that the conditions are satisfied.

14. 1. 2. 3.