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Mathematics Dictionary
Subject : Application of prgram ABH on internat
Theme : Limit[sin(x)/x] = 1 as x goes to 0
Read following questions before using Program ABH
    * Q1. Read Mathematical Symbols find symbol defintions

    * Q2. How to use the program ABH to find answer from graphs ?
      1. Click start
      2. Click a subject in upper box
      3. Click a program in lower box
      4. Click Back command and repeat step 3 or steps 2 and 3

    * Q3. How to run program 01 03 ?
      1. Start the program AGH
      2. Click Subject 01 in upper box
      3. Click program 03 in lower box

    * Q4. How to start the program ?
      1. Click Program ABH to start
      2. Select run at current location (Donot download)
      3. Select yes to run
Purposes : Use graph to verify the solution

Q1 Prove that Limit of sin(x)/x = 1 when x goes to 0

    A1. Graphic answer :
      * Start Program ABH Click here
      * Click run at current location
      * Click yes then enter program ABG
      * Click start and subjects loaded into upper box
      * Click subject 08 and programs loaded into lower box
      * Click program 01 to see graphs
      * Click back command to see other programs

    A1. Geometrical proof :
      * Picture Mathematics by Dr K. G. Shih
      * Page 166-167

    A1. Sine series method
      * Sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ....
      * Sin(x)/x = 1 - x^2/3! + x^4/5! - .....
      * Hence sin(x)/x = 1 if x=0

    A1. l'Hospital Rule
      * Limit[F(x)/G(x)] = Limit[F'(x)/G'(x)] if F(x)/G(x) = 0/0
      * sin(x)/x give 0/0
      * Let F(x)=sin(x) and F'(x)=cos(x)
      * Let G(x)=x and G'(x)=1
      * Hence Limit[sin(x)/x]=1 as x goes to zero

Definition : Lim[a*sin(x)/x] = a*Lim[sin(x)/x] = a

Definition : Lim[sin(a*x)/x] ≠ a*Lim[sin(x)/x] since sin(a*x) ≠ a*sin(x)


Q2. Limit[sin(a*x)/(b*x)] = ? when x = 0

A2. The solution is
    1. Limit[sin(a*x)/(b*x)] = Limit[(sin(a*x)/(a*x)*(a/b))
    2. = (a/b)*Limit[sin(a*x)/(a*x)]
    3. = (a/b)

Q3 Find graphic solution of Limit[sin(2*x)/x] as x=0

A3. Software ABH program 08 03
    * Click start in software ABH
    * Click subject 08 Limit
    * Click program 03 Limit[sin(a*x)/x]
    * Give input a = 2

    * Use ABH program 08 03 to Verify the identities
    * Lim[sin(2*x)/x] = 2
    * Lim[sin(3*x)/x] = 3
    * Lim[sin(4*x)/x] = 4

Q4 How to prove that Limit[sin(2*x)/x] = 2 as x=0

A4. Software ABH program 08 03
    * Graphic solution Click start in software ABH
    * Click subject 08 Limit
    * Click program 03 Limit[sin(a*x)/x]
    * Give input a = 2

    * Mathematical proof
    * Lim[sin(2*x)/x] = Lim[2*sin(2*x)/(2*x)]
    * = 2*Lim[sin(2*x)/(2*x)] = 2

    * Note : The following method is wrong but answer is right
    * Lim[sin(2*x)/x] = Lim[2*sin(x)/x]
    * = 2*Lim[sin(x)/x] = 2
    * Why wrong ? sin(2*x) ≠ 2*sin(x)

Q5. In l'hospital rule, if F'(x)/G'(x) = 0/0 what should we do ?

A4. We use Lim[F"(x)/G"(x)]
Reference

    l'Hospital Rule : Mathematics Dictionary chapter 45

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