1 / 9

L’Hospital’s Rule, Growth, and Dominance

L’Hospital’s Rule, Growth, and Dominance. Consider the following limit What do we get when we evaluate it? We can use local linearity Let’s look at the graphs. Here is the plot of . Indeterminate Forms 0/0. L’Hopital’s Rule If f and g are differentiable, f ( a ) =g ( a ) =0, then

thais
Download Presentation

L’Hospital’s Rule, Growth, and Dominance

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. L’Hospital’s Rule, Growth, and Dominance

  2. Consider the following limit • What do we get when we evaluate it? • We can use local linearity • Let’s look at the graphs

  3. Here is the plot of

  4. Indeterminate Forms 0/0 L’Hopital’s Rule If f and g are differentiable, f(a)=g(a)=0, then This can be shown by using the functions’ tangent line approximations

  5. Indeterminate Forms∞/∞ • In this case we may want to know which goes to infinity faster, the numerator or the denominator? Or do they go at about the same rate? • This can be shown using the indeterminate form 0/0

  6. L’Hopital’s Rule • If f and g are differentiable, - When limf(a)=g( , and • or • When g’(a)≠0, t hen • It can be shown that • Where a may be ±∞ • Provided the limit on the right exists

  7. Indeterminate Forms • There are other ‘indeterminate forms’ • Each of these can be changed to be 0/0 or ∞/∞ • Examples

  8. Calculate the following limits

  9. We say that g dominates f as x→∞ if • Check that x½ dominates lnx as x→∞ • Check that 2x dominates x2 as x→∞

More Related