Math 221 L’Hôpital’s Rule Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

L’Hôpital’s Rule is a tactic for finding certain limits by using derivatives. See section 4.8 in our textbook for a description and examples of the rule. This discussion gives you some experience using L’Hôpital’s Rule.

See if you can use L’Hôpital’s Rule to find...

1. \[\lim_{x \to 0} \frac{e^x-1}{x}\]
2. \[\lim_{x \to \infty} \frac{x}{e^x}\]
3. \[\lim_{x \to \infty} \frac{x}{\ln x}\]
4. \[\lim_{x \to \pi} \frac{x^2 - \pi x}{\sin x}\]

If you evaluate

without using L’Hôpital’s Rule, what do you get? If you apply L’Hôpital’s Rule to that limit, what do you get? I expect the answers to differ — can you explain why, and which one is right?