giáo án tiếng anh chuyên ngành toán - giới hạn hàm số

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

LESSON PLAN

Prepared by: Duong Lan Phuong, Math Student of Thai Nguyen University

of Education

I Objectives:

1 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

2 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 ;

Time Teacher and Students’ activities Contents

1 Definition of finite limit of a function at a point 15

mins

T: State problem.

Problem: Consider the function

2

( )

1

f x

x





1 Given value ≠1 to variable x to form a sequence

( ) xn , x n 1 as shown in the table below:

x x1=2 x2=3

2 x3=4

3 x4=5

4… x n=n+1

n … →1

f(x) f (x1) f (x2) f (x3) f (x4) … f (x n) … →1

Then, the corresponding values of the function

f(x1), f(x2), … , f (x n),… aslo form a sequence denoted

by (f(x n))

Q:

a) Prove f(x n)=2 xn=2 n+2

n b) Find the limit of sequence (f(x n))

2 Prove that for an arbitrary sequence (x n), x n ≠ 1 and

x n → 1, we always have f (x n)→2

S: Find the solution to the problem

I Finite limit of a function at a point:

T: Through this problem led the students to the

definition of finite limit of a function at a point

1 Definition 1: (textbook-p 124)

Notation: 0

lim ( )

or ( )f x  L

as x x0

T: Give example 1.

T: In the case of f ( x )= x

2

−16

x−4 (shown below) we can use a limit to find what value ƒ(x) tends to as x

tends to four

But if we substitute x = 4 into the limit, we will get

* Example 1:

Give the function f ( x )= x

2

−16

x−4 Calculate limx→ 4 f ( x )?

0 lim ( )

0

x f x

, but we know that this is wrong

Q: So how can we solve this algebraically?

T: Let’s see if we can simplify ƒ(x) at all:

2

16 ( )

4

x

f x

x





 (the numerator is a difference of two squence, so we can factorise.)

( 4)( 4)

(x 4)

x x



 (The (x – 4) in the numerator and

denominator will cancel.)

(x 4)

T: Now let’s solve for the limit of ƒ(x).

S: Solve the limit.

T: Note that if substituting for the limit produces a

zero denominator, factorise and cancel first

Solution: We have:

2

x



4 lim( 4) 8



(this means that ƒ(x) tends to 8 but does not equal 8 at the x value of 4.)

Remark:

lim

x→ x0

x=x0; lim

x → x0

c=c, with c as a constant

2 Theorem on finite limits 15

mins

S: Acknowledge the theorem.

T: Note that when we compute the limit, we rarely

use the definition that we use theorem 1 combined

with the simple limits previously known to

compute

2 Theorem on finite limits:

a Theorem 1: (textbook-p125)

T: Give example 2 Deliver the 2st worksheet to the

students

S: Work in pair to finish the task in 3 mins.

S: Applying theorem 1 to solve.

b Example 2:

a Calculate

2

1 lim ( ) lim

2

x

f x

x





b Calculate

2

2 lim ( ) lim

1

g x

x

 



T: Correct the answer.

S: Check result.

Result:

a

x x

b

2

x



1

lim( 2) 1 2 3



3 One-Sided Limits 10

mins

T: Explain the definition 2 to students.

- A one-sided limit only considers values of x 0

function that approach x value from either above 0

or below

+ The right side limit of x function f as it 0

approaches x is the limit 0 0

lim ( )

+ The left side limit of x function is 0 0

lim ( )





S: Acquire knowledge.

3 One-sided limits:

a Definition 2:

(textbook-p126)

S: Acknowledge the theorem b Theorem 2:

0

lim ( )

if and only if

lim ( ) lim ( )

T: Give example 3.

Q: - When x1,x , what is f (x)?1

- Calculate lim ( )1

 , lim ( )1

 and lim ( )1

S: Answer the question.

c Example 3: Calculate the limit

1

1

1

x

x x







Solution:

Since the absolute value function

( )

f x , is defined in a piecewise

manner, we have to consider two limits: 1

1 lim

1

x

x x







 and 1

1 lim

1

x

x x







 For x1, x 1  x 1. So

For x1, x 1 x1 So

So the two-sided limit 1

1 lim

1

x

x x





 does not exist

5 Summary the lesson 4

mins

- Review the terms learned during the lesson through flashcards

- Summary the knowledge focus

6 Homework 1

mins

- Exercises 1,2 (textbox-p132)