Find the Error: L’Hopital’s Rule

It is a beautiful Spring day. You are again jolly, for you have a backpack filled with schoolbooks and cornbread, and you are ready to have your lunch and start working on your calculus homework. You lay out your food and notebook on a picnic table, and suddenly a familiar scent catches your attention. “L’Hopitals rule, eh?” comments a wild eyed stranger who has sat next to you. “Interesting stuff… if you don’t mind reading LIES!”

“But L’Hopitals rule is useful! It is a true and wonderful technique of computing limits,” you say.

“Useless!” he bellows. “You show me a steak and baked potato with a side of L’Hopital’s rule and I’ll show you a steak and baked potato, by cracky!”

“What are you talking about?”

“Look at this, and see the lies exposed!”

The stranger writes this in your notebook:

“Why, yes, that is a very good use of L’Hopital, and it proves my point,” you say. You started with a hard limit in one of the standard indeterminate forms, used L’Hopital twice, correctly I might add, and arrived at the answer, that  does not exist.”

At that point the stranger laughs and laughs, and winds up saying between chortles, “Graph it then, Doctor Curie.”

Fearlessly, you graph (x2 + sinx)/x2, and come up with this.

“Looks like the limit is 1 to me. And last thing I heard was that 1 exists. Enjoy your johnnycake!”

You start to lose your appetite. Was L’Hopital lying all this time? Is the limit one? Or doesn’t it exist? If “one” doesn’t exist, then don’t things start looking grim for 2, 3, and 4? Does it still take one to know one? Or is there a chance that the stranger made a mistake? Find the error.

  1. Find the error: Differentiation
  2. Find the error: L’Hopital’s Rule
  3. Find the error: Related Rates
  4. Find the error: Fundamental Theorem of Calculus
  5. Find the error: U-substitution
  6. Find the error: Integration by Parts (part 1)
  7. Find the error: Integration by Parts (part 2)
  8. Find the error: Trigonometric Integration
  9. Find the error: Taylor Series
  10. Find the error: Improper Integrals and Taylor Series
  11. Find the error: Separation of Variables
  12. Find the error: Separation of Variables – Exponential Growth
  13. Find the error: Solutions
  14. Find the error: Credits