Remember solving indeterminant limits (using factoring, conjugating, etc.)? Well L’Hopital’s Rule is another method which may be used. Recall..Indeterminant forms are: Watch out! This rule may only be used if your limit is in the form and you may have to re-write it to get it that way. S. Evans

If a rational function has an indeterminant limit, You can differentiate the numerator and denominator without altering the limit value. Rule: If = Then S. Evans L’Hopital’s Rule

Ex. 1: Using l’Hopital’s rule, find: Note: if you sub -2 in you get so you may use the rule. S. Evans L’Hopital’s Rule

Ex. 2: Using l’Hopital’s rule, find: Note: if you sub 0 in you get so you may use the rule. S. Evans L’Hopital’s Rule

Ex. 3: Find: Note: if you sub in you get so you MUST rewrite as a quotient before using the rule. Also in this solution you will find you have to apply the rule three times (see next slide). S. Evans L’Hopital’s Rule

Ex. 3: S. Evans L’Hopital’s Rule

Ex. 4: Find: TRICK… Note: if you sub -2 you do NOT get an indeterminant result so you CANNOT use the rule! S. Evans L’Hopital’s Rule

CHALLENGE Ex. 5: Find: Note: if you sub in you get so you MUST rewrite as a quotient before using the rule. If expressions in terms of x are raised to a powers in terms of x, it is often useful to use natural logs to rewrite the expression. Start by setting the limit’s expression equal to y…… Let then take the ln of both sides….. S. Evans L’Hopital’s Rule

CHALLENGE Ex. 5: Since you may now rewrite the problem in terms of a quotient and apply the rule. S. Evans L’Hopital’s Rule

CHALLENGE Ex. 5: But we want the limit of y not lny! S. Evans L’Hopital’s Rule