Karl's Calculus Tutor - Solution to Exercise 8.1-2

Solution to Exercise 8.1-2KCT logo

© 1998 by Karl Hahn

The problem was, and I quote: Show how L'Hopital's Rule produces the expected result when applied to 

           f(x) - f(c)
    lim               
   x  > c    (x - c)
Assume that f(x) is continuous and differentiable over some open interval that includes x = c, and that f'(x) is continuous there as well.

To make any sense of this, you first have to determine what the expected result is. What is special about this limit? Perhaps you have already seen through its disguise. If you make the substitution of h = x - c, then x = c + h. So this limit becomes 

                f(c + h) - f(c)
        lim                    
     c+h  > c          h
But having c + h approach c is precisely the same as having h approach zero. So you really have 
            f(c + h) - f(c)
    lim                    
   h  > 0          h
Recognize it now?? This is just the limit for taking the derivative of f(x) at x = c. So we expect the limit here to be f'(c).

So now, let's apply L'Hopital to the original expression: 

           f(x) - f(c)
    lim               
   x  > c    (x - c)
Step 1: Determine that both numerator and denominator go to zero in the limit. Since f(x) is continuous, as x approaches c, it must be the case that f(x) approaches f(c). So f(x) - f(c) must approach zero. And clearly x - c goes zero as x approaches c. So this does qualify for L'Hopital's Rule.

Step 2: Take the derivatives of the numerator and denominator. But take the derivatives with respect to what? Since it is x that is the variable that approaches something, we take the derivatives with respect to x. That means that c is a constant, and hence so is f(c). The derivative of a constant is always zero. So the derivative of the numerator is f'(x) and the derivative of the denominator is 1. So we are left with 

           f(x) - f(c)              f'(x)
    lim                 =   lim          
   x  > c    (x - c)       x  > c     1

Step 3: Take the limit. The limit of f'(x)/1 as x approaches c is simply f'(c) (since f'(x) was stipulated to be continuous at x = c). So the result is as we expected it to be.

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