L'Hopital's Rule

L'Hopital's Rule

If   lim      f(x)   =    ¥      OR      lim        f(x)    =    0
     x®a   g(x)         ¥                   x®a   g(x)          0

Then   lim       f(x)    =   lim       f '(x)
          x®a   g(x)          x®a  g '(x)

Local Linearity & L'Hopital's Rule

Consider two linear functions with the same x-intercept. That is, if a is the x-intercept, the function should be of the form f(x) = m1(x - a) and g(x) = m2(x - a).
  1. What is the   lim       f(x)    ?
    2.                     x®a   g(x)

    Answer:  lim     f(x)     =     m1(x - a)    =    m1
                       m2    g(x)            m2(x - a)             m2

  2. Relating this concept to non-linear functions, now consider any two differentiable functions f(x) and g(x) such that f(a) = g(a) = 0 and g '(a) is not 0 for some value, a. Name a pair of functions f and g and a value of a which satisfy these conditions.

    3. Answer:   One example
                   f(x) = (x - 4)(x + 2)
                   g(x) = (x - 4)(x2 + 1)                  a = 4

  3. Graph y = f(x) and y = g(x) on your graphing calculator and zoom in at x = 4 until f and g appear linear.
  4. Write the equations of the "lines" that approximate f and g near 4 in the form y = m(x - a).

    5.  f(x):   y = 6(x - 4)
    g(x):   y = 17(x - 4)
  5. Use these equations to compute lim      f(x)
    6.                                                  x®4  g(x)
    lim      f(x)    =   lim        6(x - 4)   =   6
    x®4  g(x)         x®4   17(x - 4)       17
  6. Check this answer against the one using the original f(x) and g(x) to calculate the limit.
  7. The slopes of the lines in #4 are the derivatives of the functions in #1 since they are the slopes of the tangent lines to the functions at x = 4.  Thus, this exercise suggests L'Hopital's Rule works.

Problems

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