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Lesson 1-b

# Lesson 1-b - Math 1314 Lesson 1 Limits What is calculus The...

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Math 1314 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. How can we find the area of a region bounded by an arbitrary curve?

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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 :
It does not matter whether or not the x value every reaches the target number. It might, or it might not! Example 2 : 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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Case 2 : At the target number for the x -value, the graph of the function has a vertical asymptote. For either of these two cases, we would say that the limit as x approaches the target number “does not exist.” Definition : We say that a function f has limit L as x approaches the target number a , written L x f a x = ) ( lim if the value f ( x
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