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Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
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Calculus I
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Calculus II
Transcendental Functions
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Calculus III

Course: Calculus II
Topic: Calculus of Infinity
Subtopic: Indeterminate Forms and L'Hopital's Rule

Overview

We know that zero divided by a non-zero number is zero, and that a non-zero number divided by zero is "undefined", but what about zero divided by zero? We know that 0^1=0 and that 1^0=1, but what about 0^0? Things like 0/0 or 0^0 are called indeterminate forms because they are not necessarily determinable. We will look at them in terms of limits, such as lim(x/sin(x)) as x->0 (which appears to approach 0/0).

Early in Calculus I we studied various approaches to evaluating limits (worth reviewing now!) including those of the form 0/0. We attacked them using algebra to rewrite or simplify the function. But a nice alternative is to use L'Hopital's Rule. This rule can be used to evaluate limits of the form 0/0 or ∞/∞.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 9.1.1 Recognize the four "special limits" from Calculus I and how to evaluate them
• 9.1.2 Recognize determinate forms and indeterminate forms
• 9.1.3 Know the seven indeterminate forms and know not to make assumptions about their results
• 9.1.4 Understand when L'Hopital's Rule applies to a limit and when it does not
• 9.1.5 Be able to evaluate a limit using L'Hopital's Rule once or repeatedly
• 9.1.6 Be able to transform an indeterminate form so that L'Hopital's Rule applies
• 9.1.7 Evaluate a limit of a variable expression to a variable expression either by letting the limit equal y and applying the natural logarithm to both sides or rewriting x as e^lnx to transform an indeterminate form into one where LH can be used

Terminology

Terms you should be able to define: the four "special limits" from Calculus I (shown again below), determinate form, indeterminate form, L'Hopital's Rule

Special Limits

 lim_(theta->0) sintheta/theta = 1 lim_(theta->0) tantheta/theta = 1 lim_(theta->0) (1-costheta)/theta = 0 lim_(x->0) (1+x)^(1/x) = e

Mini-Lectures and Examples

Supplemental Resources (recommended)