Estimation and Limits -- Multiplying Functions -- Answers

1. $50.00. If the best pump and the best timer are used then

   3.98  < time < 4.02
   0.249 < rate < 0.251
   
   0.99102 < water < 1.00902
   

   So the result will be within the required tolerance. Any other combination will not insure the required tolerance.

2. First we show that

     Lim   h(x) = h(a)
   x --> a
   
     Lim   h(x) =  Lim  A(x) A(x)
   x --> a       x --> a 
   
                =  Lim  A(x)   Lim  A(x)
                 x --> a     x --> a
   
                =  A(a) A(a)
   
                = a^2
   
                = h(a)
   

   Now we can show that for the function v(x) = x^3

     Lim   v(x) = v(a)
   x --> a
   

   by noticing that v(x) = h(x) A(x) -- that is, that x^3 = x^2 x.

   The same argument shows that for the function s(x) = x^4 = x^3 x

     Lim   s(x) = s(a)
   x --> a
   

   and continuing in this way we see that for each power function q(x) = x^n

     Lim   q(x) = q(a)
   x --> a
   

   Now, for any function w(x) = k x^n we have

     Lim   w(x) =   Lim   k q(x)
   x --> a        x --> a
   
                = k   Lim   q(x)
                    x --> a
   
                = k q(a)
      
                = k a^n
   
                = w(a)
   

   Finally, we show that for any polynomial p(x)

     Lim   p(x) = p(a)
   x --> a
   

   by showing one-by-one that for each of the functions

   

   that