# Day-1 - Math 1314 Day 1 Notes Instructor Marjorie Marks...

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Math 1314 Day 1 Notes Instructor: Marjorie Marks Email: . CASA Website: casa.uh.edu My personal website: math.uh.edu/~mmarksc Prerequisites: You must have credit for College Algebra. You can earn credit by placing out of Math 1310 on the UH Math Placement Test or by successfully completing College Algebra here or at another institution. Not only do you need to have credit in that class, I expect you to remember just about everything from College Algebra. There are lots of online resources to help you, but if you don’t remember much of College Algebra, this class may not be for you. Go to my personal website, www.math.uh.edu/~mmarksc . Click on the link to Math 1314. There’s a link to “Review Materials.” On the resulting page, there’s a link to Prerequisite Review. There are worked problems and flash videos of review problems. You need to know virtually everything on that review.

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Section 10.4 - 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?
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 Informal definition: 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. Look at these graphs and find the limit as x gets really close to 1 in both cases. -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 x y -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 x y

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Example 1: Find 2 lim ( ) x f x . Find 0 lim ( ) x f x . -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4 x y It does not matter whether or not the x value every reaches the target number. It might, or it might not! Example 2 : Find 1 lim ( ) x f x . -4 -3 -2 -1 1 2 3 4 5 -4 -3 -2 -1 1 2 3 4 x y
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: 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.”

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More Formal 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 ) can be made as close to the number L as we please by taking x sufficiently close to (but not equal to) a . Note that
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Day-1 - Math 1314 Day 1 Notes Instructor Marjorie Marks...

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