Our last unit this year is limits. In basic terms, you have some function, f(x). It can be anything, from f(x)=x+1 to f(x)= x^2-1 to f(x) = (Sin(Cos(6x-10)))^3. The limit is the y value that the function approaches as x gets closer and closer to some particular value (usually denoted as c for some constant). You can also find limits of a function as the function approaches positive or negative infinity. We will work on those limits next week. For this week, we will work on functions that are approaching a specific numerical limit value (c). Let's say the function is f(x) = x+1 and you want to find the limit of f at x = 5. As x gets closer and closer to 5 (imagine plugging in values of x where x = 1, then x=2, then x=3, then x = 4, x=4.5, x=4.9, x= 4.999, etc.). The y values that result are getting closer and closer to 6. Thus the limit of f(x) = x+1 as x approaches the value of five is 6. Y is getting closer and closer to 6. For the same function, the limit of f(x) = x+1 as x approaches 99 would have a limit of 100. Limits are very useful in calculus and finding results for problems that can not be calculated directly. The above example is a simple one. Calculating a limit will often be more involved then simply plugging in the value and finding what y equals. Limits can be found graphically, numerically and algebraically. The first ways we will try are graphically and numerically. The problems for homework are: page.889: #'s 2, 8, 14, 24 and page. 897 #'s 1 and 2 These are due Monday, 6/8/20. Finding limits algebraically takes a little more work. The easiest way to find a limit algebraically is to plug the limit number into a function. If you get an answer, that value is the limit. But very often, the function is undefined at the plug in limit value. Even though a function is undefined at some value, the limit at that value may or may not exist. In these cases, you will have to do a bit more work to determine what limit if any exists. The problems for homework are: page 897 #'s 4-20 evens, 33. These are due Thursday, 6/11/20. Below are some videos giving examples on limits and calculating the limit as a function approaches a specific constant value. Introduction to limits www.youtube.com/watch?v=ahZ8LLtgu_w Properties of limits (rules for how you can combine the limits of different functions) www.youtube.com/watch?v=US9EXMXqM3I&feature=youtu.be Determining a limit numerically (no algebra. use of calculator) www.youtube.com/watch?v=l7Tcay720vw www.youtube.com/watch?v=KesTHnYwRMg www.youtube.com/watch?v=58u7vaqHg68 Determining a limit graphically. www.youtube.com/watch?v=LdewtuWi7fM www.youtube.com/watch?v=BsgrGFIbMdU&feature=youtu.be www.youtube.com/watch?v=Vi4BiJj-n0g&feature=youtu.be www.youtube.com/watch?v=fky6rtTMOwM&feature=youtu.be www.youtube.com/watch?v=3iZUK15aPE0 www.youtube.com/watch?v=-f_U7Asybsk&feature=youtu.be Finding limits Algebraically: Plugging in limit value: www.youtube.com/watch?v=VLiMfJHZIpk Algebraically when plug-in doesn't work. - Factoring and then plug in limit value www.youtube.com/watch?v=qHfyB0J57qo&feature=youtu.be www.youtube.com/watch?v=gRk24f3SUWQ Finding limits algebraically when plugging in factoring will not work www.youtube.com/watch?v=HnmWte9ZTLE Algebraically by rationalizing when plugging in doesn't work. www.youtube.com/watch?v=ptgGkxhExA0&feature=youtu.be www.youtube.com/watch?v=ouWAhqeAaik&feature=youtu.be Examples of finding different limits algebraically. (Can plug in, factor and cancel, rationalize, one sided limits) www.youtube.com/watch?v=FItPk577Shg
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