1434.8 L’Hospital’s rule: limits revisitedTextbook pages 299-307L’Hospital’s rule is a very powerful tool for finding limits of indeterminate limits such asRule for the indeterminate form00:Examples for the indeterminate form00:• Find the limit ofsin(x)xat x = 0.• Find the limit ofx2−9x−3at x = 3.• Find the limit oftan(x)xat x = 0.144 CHAPTER 4. DIFFERENTIAL CALCULUSProof for the indeterminate form00:Rule for the indeterminate form∞∞:Examples for the indeterminate form∞∞:• Find limx→+∞ln(x)x.• Find limx→(π/2)−tan(x)1+tan(x).The proof of this indeterminate form is more difficult, so we will simply assume that it is true.Advantages:• L’Hospital Rule can be applied multiple times4.8. L’HOSPITAL’S RULE: LIMITS REVISITED 145• L’Hospital Rule can also be used to find limits of other indeterminate forms providedwe can cast them into either of the above 0/0 or ∞/∞.Examples:• Find limx→0x2sin2(x).• Find limx→01sin(x)− cot(x)Important note: L’Hospital rule ONLY works for these two indeterminate forms. Do nottry to use it without verifying that the limit is one of those forms first. For example, youcannot use L’Hospital Rule on the following problems:• limx→3x2−1x−3• limx→0xln(x)• limx→∞e−xx2(and so forth)...146 CHAPTER 4. DIFFERENTIAL CALCULUSCheck your understanding of Lecture 22• Limits using L’Hospital’s RuleDo as many problems as you can from the following list: Textbook page 307 numbers 1through 32 (a good start would be all of the odd-numbered, or all of the even numberedproblems on that list).• Further examples of L’Hospital’s RuleTextbook page 308 number 51,