L’Hopital’s Rule Section 8.1a

Indeterminate Form 0/0 Consider the limit: If both functions are zero at x = a, then substitution produces the result 0/0. This is a meaningless expression that is known as an indeterminate form. Past experiences with indeterminate forms: Our new rule will enable us to use our knowledge of derivatives to evaluate limits that otherwise lead to indeterminate forms!

L’Hopital’s Rule (First Form) Start with the ratio of the derivatives of the functions: Def. of Deriv. Prop. of Limits The original limit!!! Rewrite Given info.

L’Hopital’s Rule (First Form) Suppose that , that and exist, and that . Then

L’Hopital’s Rule (First Form) Evaluate Form Look familiar?

L’Hopital’s Rule (First Form) Evaluate Form Verify graphically?

L’Hopital’s Rule (First Form) Evaluate Form Verify graphically?

L’Hopital’s Rule (Stronger Form) Sometimes after differentiation the new numerator and denominator both equal zero at x = a. In such cases we apply the stronger form: Suppose that , that and are differentiable on an open interval containing , and that on if . Then

L’Hopital’s Rule (Stronger Form) Evaluate Form Still Differentiate again… No longer indeterminate Verify graphically?

L’Hopital’s Rule (One-Sided Limits) Evaluate and Both form What is suggested from the graph of the function?

Other Indeterminate Forms The rule applies to quotients that lead to the indeterminate forms or . In some other cases, it is possible to use algebra to get to one of these forms… Evaluate Form

Other Indeterminate Forms Evaluate Form (Remember, we are not truly “substituting in infinity,” just as these forms are not truly numbers… they are simply descriptions of function behavior)

Other Indeterminate Forms Evaluate Form The graph suggests that the limit exists, so rewrite: Let

Other Indeterminate Forms Evaluate Form The graph suggests that the limit exists, so rewrite: Now form

Other Indeterminate Forms Evaluate Form Still