1 / 12

Exploring Limits and Derivatives of Logarithmic Functions

This document provides a thorough exploration of finding limits and derivatives related to logarithmic functions. It discusses various scenarios where limits exist and how to determine them, alongside different derivative applications. Through examples, the text highlights the process of differentiating logarithmic functions and illustrates the concept of e as a limit. Additionally, it covers important definitions and forms of limits, aiding in the understanding of the rules governing logarithmic differentiation. Homework exercises are also included for practice.

salma
Download Presentation

Exploring Limits and Derivatives of Logarithmic Functions

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. Question • Suppose exists, find the limit: (1) (2) Sol. (1) (2) • (1) Suppose exists and then (2) Suppose as then

  2. Question Suppose exists and find the limit The solution is Sol.

  3. Derivatives of logarithmic functions • The derivative of is • Putting a=e, we obtain

  4. Example Ex. Differentiate Sol. Ex. Differentiate Sol.

  5. Question Find if Sol. Since it follows that Thus for all

  6. Example Find if Sol. Since it follows that and by definition, Thus for all x

  7. Question Find if (a) (b) (c) Sol. (a) (b) (c)

  8. The number e as a limit • We have known that, if then • Thus, which by definition, means • Or, equivalently, we have the following important limit

  9. Other forms of the important limit • Putting u=1/x, we have • More generally, if then

  10. Question Suppose exists and find the limit The solution is Sol. Let then

  11. Question Discuss the differentiability of and find Sol. does not exist

  12. Homework 6 • Section 3.6: 46, 49, 50 • Section 3.7: 16, 20, 34, 35, 39, 40, 63 • Section 3.8: 41, 45, 48

More Related