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MATH 261 Limits and Derivatives Notes

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Lecture Notes CALCULUS I - MATH 261 Chapter 2 – Limits and Derivatives 2.1 – The Tangent and Velocity Problems 2.2 – The limit of a function 2.3 – Calculating Limits using the Limit Laws 2.4 – The Precise definition of a Limit 2.5 – Continuity 2.6 – Limits at Infinity; Horizontal Asymptotes 2.7 – Derivatives and Rates of Change 2.8 – The Derivative as a Function 1 2.2 – The Limit of a Function Important New Idea: The notion of the Limit à This idea distinguishes calculus from other branches of mathematics à In fact, we might define calculus as the study of limits Definition: (intuitive definition of limits) Left hand limit: lim 𝑓(𝑥) = 𝐿 means $→& ' as 𝑥 approaches 𝑎 from the left side, 𝑓(𝑥) approaches 𝐿 Right hand limit: lim 𝑓(𝑥) = 𝐿 means $→& - as 𝑥 approaches 𝑎 from the right side, 𝑓(𝑥) approaches 𝐿 Limit: lim 𝑓(𝑥) = 𝐿 if and only if $→& lim 𝑓(𝑥) = 𝐿 = lim- 𝑓(𝑥) $→& ' $→& Note 1: L is unique. This means that if the limit exists, it should be only one value. Note 2: If lim' 𝑓(𝑥) ≠ lim- 𝑓(𝑥), we say lim 𝑓(𝑥) Does not exist Note 3: Notion of the limit is associated with the behavior of 𝑓(𝑥 ) NEAR 𝑎, not AT 𝑎. àNothing is required at 𝑎 à𝑓 does not need to be defined at 𝑎 $→& $→& $→& 2 ∎Finding limits using a graph Example 1: Use the graph of 𝑓 (𝑥 ) provided below to find each limit. a. li ...
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