Basic Calculus SHS Q3 Lesson 1 - Evaluating Limits PDF

Precalculus /Basic Calculus Basic Calculus Science, Science, Technology, Technology, Engineering, Engineering, and Mathematics and Mathematics When you ride a car, do you notice its speed?...

Precalculus /Basic Calculus Basic Calculus Science, Science, Technology, Technology, Engineering, Engineering, and Mathematics and Mathematics When you ride a car, do you notice its speed? 2 Usually, we get the average speed of the car for a certain duration, like one hour or one minute. But is it possible to estimate the speed of the car at a particular instant? 3  To answer this question, we need the concept of limits. In particular, we will talk about limits of functions. 4  Learning Competency At the end of the lesson, you should be able to do the following: Illustrate the limit of a function using a table of values (STEM_BC11LC-IIIa-1). 5  Learning Objectives At the end of the lesson, you should be able to do the following: Define the limit of a function. Define one-sided limits. Define infinite limits. Estimate the limit of a function using tables of values. 6 Is it correct to say that we can only get an estimate of a limit? Why do you say so? 7  Limit of a Function Investigate what happens to the values of the linear function 𝒇 𝒙 = 𝒙 + 𝟒 as 𝒙 approaches 2. 8  Limit of a Function from the left of 2: 𝒙𝟐 2.100 2.050 2.010 2.005 2.001 𝟐 𝒇(𝒙) 9  Limit of a Function from the left of 2: 𝒙𝟐 2.100 2.050 2.010 2.005 2.001 𝟐 𝒇(𝒙) 6.100 6.050 6.010 6.005 6.001 11  Limit of a Function from the left of 2: 𝒙𝟐 2.100 2.050 2.010 2.005 2.001 𝟐 𝒇(𝒙) 6.100 6.050 6.010 6.005 6.001 12  Limit of a Function We say that “the limit of 𝑓(𝑥) as 𝑥 approaches 2 is 6.” In symbols, lim (𝑥 + 4) = 6. 𝑥→2 13  Intuitive Definition of a Limit Suppose the function 𝑓 𝑥 is defined when 𝑥 is near 𝑐. If 𝑓 𝑥 gets closer to 𝐿 from both sides as 𝑥 gets closer to 𝑐, then we say that “the limit of 𝒇(𝒙) as 𝒙 approaches 𝒄 is equal to 𝑳.” 𝐥𝐢𝐦 𝒇 𝒙 = 𝑳 𝒙→𝑐 14  Let’s Practice! Estimate the limit of the quadratic function 𝒈 𝒙 = 𝒙𝟐 − 𝟔𝒙 + 𝟏𝟒 as 𝒙 approaches 𝟒 using tables of values. 15  Let’s Practice! Estimate the limit of the quadratic function 𝒈 𝒙 = 𝒙𝟐 − 𝟔𝒙 + 𝟏𝟒 as 𝒙 approaches 𝟒 using tables of values. 𝟔 16 Try It! Estimate the limit of the function 𝒈 𝒙 = 𝟓𝒙 + 𝟖 as 𝒙 approaches 𝟏 using tables of values. 17  Let’s Practice! Estimate 𝐥𝐢𝐦 𝒙 using table of values. 𝒙→𝟗 18  Let’s Practice! Estimate 𝐥𝐢𝐦 𝒙 using table of values. 𝒙→𝟗 𝟑 19 Try It! Estimate 𝐥𝐢𝐦 𝒙 − 𝟒 using tables of 𝒙→𝟏𝟑 values. 20  Let’s Practice! sin 𝒙 Estimate 𝐥𝐢𝐦 using table of values. 𝒙→𝟎 𝒙 21  Let’s Practice! sin 𝒙 Estimate 𝐥𝐢𝐦 using table of values. 𝒙→𝟎 𝒙 𝟏 22 Try It! 𝒙𝟐 +𝟑𝒙+𝟐 Estimate 𝐥𝐢𝐦 using tables of 𝒙→−𝟏 𝒙+𝟏 values. 23 Remember In finding the limit, we are only concerned about the value being approached by the function 𝑓(𝑥) as 𝑥 approaches a number 𝑎. The function need not be defined at 𝑥 = 𝑎. 24 Does the limit of a function always exist? 25  One-Sided Limits Left-hand Limit: Suppose the function 𝑓 𝑥 is defined when 𝑥 is near 𝑐 from the left. Then, the limit of 𝑓(𝑥) as 𝑥 approaches 𝑐 from the left is equal to a number 𝑀. This can be written as 𝐥𝐢𝐦− 𝒇 𝒙 = 𝑴. 𝒙→𝑐 26  One-Sided Limits Right-hand Limit: Suppose the function 𝑓 𝑥 is defined when 𝑥 is near 𝑐 from the right. Then, the limit of 𝑓(𝑥) as 𝑥 approaches 𝑐 from the right is equal to a number 𝑁. This can be written as 𝐥𝐢𝐦 𝒇 𝒙 = 𝑵. 𝒙→𝑐 + 27  One-Sided Limits If the left- and right-hand limits of a function 𝑓(𝑥) as 𝑥 approaches 𝑐 are both equal to a certain real number, then the limit of the function 𝑓(𝑥) as 𝑥 approaches 𝑐 exists. Otherwise, it does not exist. 28  Let’s Practice! Estimate the limit of the signum function 𝒔 𝒙 as 𝒙 approaches zero from the left. −𝟏, 𝒙𝟏 29  Let’s Practice! Estimate the limit of the signum function 𝒔 𝒙 as 𝒙 approaches zero from the left. −𝟏, 𝒙𝟏 –𝟏 30 Try It! Given the piecewise function 𝒕(𝒙) below, estimate 𝐥𝐢𝐦− 𝒕(𝒙) using a table of 𝒙→𝟏 values. 𝟐𝒙 + 𝟑, 𝒙