1314L1 - Lesson 1 Limits What is calculus? The body of...

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Lesson 1 – Limits 1 Lesson 1 Limits What is calculus? The body of mathematics that we call calculus resulted from the investigation of two basic questions by mathematicians in the 18 th century. 1. How can we find the line tangent to a curve at a given point on the curve? -2 -1 1 2 -1 1 x y x 2. How can we find the area of a region bounded by an arbitrary curve? The investigation of each of these questions relies on the process of finding a limit , so we’ll start by informally defining a limit and follow that by learning techniques for finding limits. Limits Finding a limit amounts to answering the following question: What is happening to the y - value of a function as the x -value approaches a specific target number? If the y -value is approaching a specific number, then we can state the limit of the function as x gets close to the target number. Example 1: Given the following graph: What is the limit of the function as x approaches -1? What is the limit of the function as x approaches 0? -2 -1 1 2 1 x y
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Lesson 1 – Limits 2 It does not matter whether or not the x value ever reaches the target number. It might, or it might not! Example 2: Given the following graph, what is the limit of the function as x approaches 0? -2 -1 1 2 -1 1 x y Example 3: Given the following function, what is the limit of the function as x approaches 1? g G g ± ² > + - = < = 1 x if , 2 1 x if , 0 1 x if , ) ( x x x f When can a limit fail to exist? We will look at two cases where a limit fails to exist (note: there are more, but some are beyond the scope of this course). Case 1 : The y value approaches one number from numbers smaller than the target number and it approaches a second number from numbers larger than the target number:
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Lesson 1 – Limits 3 Case 2 : At the target number for the
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1314L1 - Lesson 1 Limits What is calculus? The body of...

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