Limit Of A Function Dương Lan Phương TO50B LESSON PLAN Prepared by: Duong Lan Phuong, Math Student of Thai Nguyen University of Education Grade: 11 Subject: Mathematics Chapter: IV Lesson: Lesson Title: LIMIT OF A FUNCTION I Objectives: Lesson objectives: - By the end of this lesson, the student will be able to compute a basic limit of a function using limit notation - The student will have an intuitive understanding of the limiting process; the ability to calculate limits using algebra, and will be able to estimate limits from graphs or tables of data Language objective: - Listen terms of limits of function and their definition - Use these terms in problems - Present solutions to get used to using new terms II Subject Matter: - Reference: Algebra & Analytics 11 Textbook - Materials: Sheets of paper, ruler,… III Procedure: Outline of the lesson: - Study two first sections of the topic - Learn to know about finite limit of a function at a point, theorem on finite limits and one-sided limits through some examples - Practice doing problems Notation: T: teacher ; S: Student ; Q: questions; Ans: Answer ; Limit Of A Function Time Dương Lan Phương TO50B Teacher and Students’ activities Contents Definition of finite limit of a function at a point 15 mins T: State problem f (x ) = Problem: Consider the function I Finite limit of a function at a point: 2x − 2x x −1 Given value ≠1 to variable x to form a sequence (xn ) xn → , as shown in the table below: x … … →1 f(x) … … →1 Then, the corresponding values of the function aslo form a sequence denoted by Q: a) Prove b) Find the limit of sequence Prove that for an arbitrary sequence and , we always have S: Find the solution to the problem T: Through this problem led the students to the definition of finite limit of a function at a point Definition 1: (textbook-p 124) lim f ( x) = L x → x0 Notation: as x → x0 f (x) → L or T: Give example * Example 1: T: In the case of (shown below) we can use a limit to find what value ƒ(x) tends to as x tends to four Give the function Calculate ? But if we substitute x = into the limit, we will get lim f (x) = x→4 0 , but we know that this is wrong Q: So how can we solve this algebraically? Limit Of A Function Dương Lan Phương TO50B T: Let’s see if we can simplify ƒ(x) at all: x − 16 f (x ) = x−4 = (x + 4)(x − 4) (x− 4) = (x + 4) (the numerator is a difference of two squence, so we can factorise.) (The (x – 4) in the numerator and denominator will cancel.) T: Now let’s solve for the limit of ƒ(x) Solution: We have: S: Solve the limit T: Note that if substituting for the limit produces a zero denominator, factorise and cancel first x − 16 (x + 4)(x − 4) = lim x→4 x − x →3 (x− 4) lim = lim( x + 4) = x →4 (this means that ƒ(x) tends to but does not equal at the x value of 4.) Remark: , with c as a constant Theorem on finite limits 15 mins S: Acknowledge the theorem Theorem on finite limits: T: Note that when we compute the limit, we rarely use the definition that we use theorem combined with the simple limits previously known to compute a Theorem 1: (textbook-p125) T: Give example Deliver the 2st worksheet to the students b Example 2: S: Work in pair to finish the task in mins a Calculate S: Applying theorem to solve b Calculate T: Correct the answer Result: x2 + lim f (x) = lim x →3 x →3 x x2 + x − limg (x) = lim x →1 x →3 x −1 Limit Of A Function Dương Lan Phương TO50B S: Check result a b x2 + 32 + lim = lim = x →3 x x →3 3 x2 + x − (x − 1)(x + 2) lim = lim x →1 x → x −1 (x− 1) = lim(x + 2) = 1+ = x →1 One-Sided Limits 10 mins T: Explain the definition to students One-sided limits: - A one-sided limit only considers values of x0 function that approach or below + The right side limit of approaches x0 x0 a Definition 2: (textbook-p126) value from either above x0 function f as it lim f (x) = L is the limit x0 + The left side limit of x → x0+ lim f (x ) = L function is x → x0− S: Acquire knowledge S: Acknowledge the theorem b Theorem 2: lim f (x) = L x → x0 if and only if lim− f (x) = lim+ f (x) = L x → x0 T: Give example c Example 3: Calculate the limit x −1 lim x →1 x − x > 1, x < Q: - When , what is f (x)? lim f (x) lim+ f (x) - Calculate x →1− , x →1 lim f (x) and x → x0 x →1 ? Solution: Limit Of A Function Dương Lan Phương TO50B S: Answer the question Since the absolute value function f (x) , is defined in a piecewise manner, we have to consider two x −1 x −1 lim− lim+ x →1 x − x →1 x − limits: and x > 1, x − = x − For So lim+ x →1 x −1 x −1 = lim+ x →1 x −1 = x −1 x < 1, x − = − x − For So lim− x →1 x −1 x −1 = lim− = −1 x − x→1 − x − lim x →1 So the two-sided limit does not exist Summary the lesson mins - Review the terms learned during the lesson through flashcards - Summary the knowledge focus Homework mins - Exercises 1,2 (textbox-p132) x −1 x −1 ...Limit Of A Function Time Dương Lan Phương TO50B Teacher and Students’ activities Contents Definition of finite limit of a function... know that this is wrong Q: So how can we solve this algebraically? Limit Of A Function Dương Lan Phương TO50B T: Let’s see if we can simplify ƒ(x) at all: x − 16 f (x ) = x−4 = (x + 4)(x − 4)... lim f (x) = lim x →3 x →3 x x2 + x − limg (x) = lim x →1 x →3 x −1 Limit Of A Function Dương Lan Phương TO50B S: Check result a b x2 + 32 + lim = lim = x →3 x x →3 3 x2 + x − (x − 1)(x + 2)