121616949-math.97

# 121616949-math.97 - respectively zero 1 and ∞ We can in...

This preview shows page 1. Sign up to view the full content.

4.8 Limits revisited 83 EXAMPLE 4.14 Compute lim x 0 sec x - 1 sin x . Both the numerator and denominator approach zero, so applying L’Hˆopital’s Rule: lim x 0 sec x - 1 sin x = lim x 0 sec x tan x cos x = 1 · 0 1 = 0 . EXAMPLE 4.15 Compute lim x 0 + x ln x . This doesn’t appear to be suitable for L’Hˆopital’s Rule, but it also is not “obvious”. As x approaches zero, ln x goes to -∞ , so the product looks like (something very small) · (something very large and negative). But this could be anything: it depends on how small and how large . For example, consider ( x 2 )(1 /x ), ( x )(1 /x ), and ( x )(1 /x 2 ). As x approaches zero, each of these is (something very small) · (something very large), yet the limits are
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: respectively zero, 1, and ∞ . We can in fact turn this into a L’Hˆopital’s Rule problem: x ln x = ln x 1 /x = ln x x-1 . Now as x approaches zero, both the numerator and denominator approach inﬁnity (one-∞ and one + ∞ , but only the size is important). Using L’Hˆopital’s Rule: lim x → + ln x x-1 = lim x → + 1 /x-x-2 = lim x → + 1 x (-x 2 ) = lim x → +-x = 0 . One way to interpret this is that since lim x → + x ln x = 0, the x approaches zero much faster than the ln x approaches-∞ ....
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern