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UNLV - MATH 181 - Class Notes - Week 2

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UNLV - MATH 181 - Class Notes - Week 2

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background image Key Idea: What distinguishes Calculus from other subjects (e.g. Algebra) is the concept of the  limit.
Limits are based on functions
Calculus therefore is defined by functions and limits. Definition: Intuitive - If the value of f(x) approaches the number L as x approaches a from the 
left, we write:
lim 𝑓(𝑥) = 𝐿 𝒙 𝒇(𝒙) = 𝒚 𝑥 𝑓(𝑥 ) = 𝑦 𝑥 𝑓(𝑥 ) = 𝑦 𝑥 𝑓(𝑥 ) = 𝑦 --- --- a L 𝑓(𝑥 ) = 𝑦
𝑓(𝑥 ) = 𝑦
𝑓(𝑥 ) = 𝑦
𝑥        𝑥       𝑥 Example: Here we have one-sided limits, which is better than nothing,  but we still have a jump… lim 𝑓(𝑥) = 1 lim 𝑓(𝑥) = −1 1 Example: Better…the one-sided limits are the same, but no jumps lim 𝑓(𝑥) = 1 lim 𝑓(𝑥) = −1 …in which case, according to the definition of the limit lim 𝑓(𝑥) = 0 = lim 𝑓(𝑥) or lim 𝑓(𝑥) = − in which case lim 𝑓(𝑥) = 𝑓(0) Definition: A function f is continuous at x = a provided lim 𝑓(𝑥) = 𝑓(𝑎) F is defined at x = a: that is, 𝑎 ∈ D(f) (which means that f(a) is a number). 1. The limit exits 2. The limit equals the value of the function 3. Note: The definition is misleading. We must check 3 things: Polynomials at any point 1. Rational functions 2. Trigonometric functions 3. Exponential functions 4. Note: What types of functions are continuous? These: Example: 𝑎 ∈ 𝐷(𝑓). Let 𝑓(𝑥) = 2𝑥 − 3𝑥 + 4 lim 𝑓(𝑥) where a = 1 Evaluate limit using point continuity at 𝑎 = 1 ∈ 𝐷(𝑓) 1. lim 𝑓(𝑥) = 𝑓(1) = 3 Where
𝑓(1) = 2(1) − 3(1) + 4 = 3
𝑓 𝑓(𝑥) = 2𝑥 − 3𝑥 + 4 𝑋 = 𝐷(𝑓) = ℝ 𝑌 = ℝ Note: It is very misleading to write the equation as: lim 𝑥 − 1 𝑥 − 1 ⎯⎯⎯⎯⎯⎯= lim  𝑥 + 1 = (1) + 1 = 2 since these are two different functions! Example: Let 𝑓(𝑥) = 𝑥 + 1    𝑥 < 1 −2𝑥 + 4    𝑥 ≥ 1 Note: 𝑓 𝑥 = 𝑥 + 1    𝑥 < 1 −2𝑥 + 4    𝑥 ≥ 1 𝑋 = 𝐷 𝑓 = ℝ 1 f 𝒈 2. Limits and Continuity Monday, January 28, 2019 10:34 AM     Chapter 1 - Functions and Limits Page 1    

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School: University of Nevada - Las Vegas
Department: Applied Mathematics
Course: Calculus I
Professor: Michael Marcozzi
Term: Spring 2019
Tags: Math181 and Calculus
Name: MATH 181 Week 2 Notes
Description: These cover the second section of Chapter 1.
Uploaded: 01/30/2019
2 Pages 54 Views 43 Unlocks
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