Counter
Mathematics Dictionary
Dr. K. G. Shih
Limit and Summation
Subjects


  • CA 01 00 | - Outlines
  • CA 01 01 | - Lim[(F(x+h) - F(x))/h] as h goes to zero
  • CA 01 02 | - Lim[sin(x)/x] as x goes to zero
  • CA 01 03 | - Lim[(1 + x)^(1/x)] = e as x goes to zero
  • CA 01 04 | - Lim[(1 + 1/x)^x] = e as x goes to infinite
  • CA 01 05 | - Lim[(Exp(x) - 1)/x] = 1 as x goes to zero
  • CA 01 06 | - Lim[tan(x)/x] as x goes to zero
  • CA 01 07 | - Lim[cos(x)/x] as x goes to zero or infinite
  • CA 01 08 | - Summation
  • CA 01 09 | - Area under curve y = 1/(1 + x^2) = pi/4 for x = 0 to x = 1
  • CA 01 10 | - New
    • Answers


    CA 01 01. Lim[(F(x+h) - F(x))/h] as h goes to zero

    Definition
    • Study Subject | Lim[(F(x+h)-F(x))/h] = 1 as x goes to zero
    • Limit is used to find the derivative
    • Two points P(x1,y1) and Q(x2,y2) : (y2 - y1)/(x2 - x1) is slope of line PQ
    • Let y = F(x), x2 = x + h, y1 = F(x), y2 = F(x + h) and x2 - x1 = h
    • (F(x+h)-F(x))/h = slope of PQ
    • When h goes to zero, slope of PQ is the slope of tangent at P
    Reference
    • Derivative and slope - Program 02 01

    Go to Begin

    CA 01 02. Lim[sin(x)/x] as x goes to zero

    Topic
     Outline
    • Diiferent proof are given in above web page
    • It is used to find derivative of sin(x)
    • d/dx(sin(x)) is given in program 03 01

    Go to Begin

    CA 01 03. Lim[(1+x)^(1/x)] = e

    Topic
    Outline
    • It is used to find derivative of ln(x)
    • d/dx(ln(x)) = 1/x is given in program 06 01

    Go to Begin

    CA 01 04. Lim[(1 + 1/x)^(x)] = e

    Topic
     Outline
    • Use binomial theory to prove this limit
    • Then wan prove that Lim[(1+x)^(1/x)] = e as x goes to zero

    Go to Begin

    CA 01 05. Lim[(Exp(x) - 1)/x] = 1 as x goes to zerp
    Outline
    • It is used to find derivative of d/dx(exp(x)) = exp(x) in program 05 01
    Proof
    • Since e^x = 1 + x + (x^2)/2! + ...
    • Hence Lim[(1 + x + (x^2)/2! + .... - 1)/x]
    • Hence Lim[(x + (x^2)/2! + ....)/x]
    • Hence Lim[(1 + (x)/2! + ....)] = 1 as x = 0

    Go to Begin

    CA 01 06. Lim[tan(x)/x] = 1 as x goes to zero
    Proof
    • Since tan(x) = sin(x)/cos(x) and cos(0) = 1
    • Hence Lim[tan(x)/x] = Lim[sin(x)/x] = 1 as x = 0

    Go to Begin

    CA 01 07. Lim[cos(x)/x] = Infinite as x goes to zero
    Proof
    • Since cos(0) = 1
    • Hence Lim[cos(x)/x] = Lim[1/x] = infinite as x = 0
    Prove that Lim[cos(x)/x] = 0 as x goes to infinite
    • Since cos(x) is beween -1 and 1
    • Hence -1/x or 1/x will be zero if x goes to infinte

    Go to Begin

    CA 01 08. Summation

    Application
    • The use of derivative is to find slope of tangent to curve at a point
    • The use of summation is to find area of curve bound with x-axis
    Example : y = F(x) = 1/(1 + x^2)
    • The area bounds by the graph of y = 1/(1 + x^2) with x-axis.
    • The area from x = 0 to x = infinite is pi/2
    • The approximate solution is Sum[F(n*h)*h] = pi/2 for x = n*h (n large and h small)
    Example : Area = ∫(1/(1 + x^2))dx = arctan(x)
    • Use binomial theory 1/(1+x^2) = 1 - x^2 + (-1)*(-2)*x^4/2! + ....
    • Area
    • = ∫(1/(1 + x^2))dx
    • = ∫(1 - x^2 + (-1)*(-2)*x^4/2! + ....)dx
    • = x - (x^3)/3 + (x^5)/5 - (x^7/7) + ....
    • = arctan(x)
    Graphic solution
    • Draw the curve
    • Pick equal distance h along x-axis and draw n rectangles
    • Area of each rectangle is F(n*h)*h
    • Total area is Sum[F(n*h)*h]
    • If n is very large number and h is very small, we can get approximate answer
    Binomial theory
    Go to Begin

    CA 01 09. Area under curve y = 1/(1 + x^2) = pi/4 for x = 0 to x = 1

    Trapzoid Formula
    • Let y = F(x) and area bounded from x = x1 to x = x2
    • Area = Sum[(F(xn + h) - F(xn))*h/2]
    • Where xn = x1 + n*h and h = (x2 - x1)/h
    Two trapzoids approximation : h = 0.5
    • Trapzoid 1 :
      • x0 = 0.0 and y0 = 1.0
      • x1 = 0.5 and y1 = 1/1.25 = 0.8
      • Area = (y0 + y1)*(x1 - x0)/2 = 1.8*0.5/2 = 0.45
    • Trapzoid 2 :
      • x1 = 0.5 and y1 = 1.25
      • x2 = 1.0 and y2 = 1/2 = 0.5
      • Area = (y1 + y1)*(x2 - x1)/2 = 1.75*0.5/2 = 0.4375
    • Hence total area is 0.8875
    • The value of pi = 4*area = 3.55
    Ten trapzoids approximation : h = 0.1
    • Study Subject | Program CA 21 01
    • Form the program : What is the approximate value of pi ?
    Hundred trapzoids approximation : h = 0.01
    • Study Subject | Program CA 21 01
    • Form the program : What is the approximate value of pi ?
    Use calculator
    • Find arctan(1)
    • Find 4*arctan(1)

    Go to Begin

    CA 01 10. Answer


    Go to Begin

    CA 01 00. Outlines

    Limit : Related with derivative
    • 1. Lim[(F(x+h) - F(x))/h] = F'(x) as h goes to zero
    • 2. Lim[sin(x)/x] = 1 as x goes to zero
    • 3. Lim[(1 + x)^(1/x)] = e as x goes to zero
    • 4. Lim[(1 + 1/x)^x] = e as x goes to infinite
    • 5. Lim[(Exp(x) - 1)/x] = 1 as x goes to zero
    • 6. Lim[tan(x)/x] = 1 as x goes to zero
    • 7. Lim[cos(x)/x] = infinite as x goes to zero or infinite
    Summation : Related with integration
    • Area under curve of y = 1/(1 + x^2) = pi/4
    • Area under curve of y = 1/(1 + x^2) = ∫(1/(1 + x^2))dx = arctan(x)
    • Arctan(1) = pi/4
    Trapzoid Formula
    • Let y = F(x) and area bounded from x = x1 to x = x2
    • Area = Sum[(F(xn + h) - F(xn))*h/2]
    • Where xn = x1 + n*h and h = (x2 - x1)/h

    Go to Begin
    Show Room of MD2002 Contact Dr. Shih Math Examples Room
    Copyright © Dr. K. G. Shih, Nova Scotia, Canada.
    1