Example 8.4.9: Applying the Lagrange Remainder
f(x) =where r(x) goes to zero as x goes to c.
Use this result and the function f(x)= to show that
The first statement is a straight-forward application of the Lagrange remainder theorem - try it youself!
As for the application, let f(x)=, which is continuously differentiable around c = 0. According to our statement (and taking the first derivative at zero) we have:
= 1 + x/2 + x r(x)
for some r(x) with r(x) = 0. To apply this to our problem, we need to see involved somehow. Therefore we factor an n to get:
Now we can apply our approximation to the second root, with x = 1/, to get:
which, together with the fact that r(x) = 0 will do the trick.