## Example 8.4.9: Applying the Lagrange Remainder

Show that if

*f*is*n*-times continuously differentiable on*[a, b]*and*c [a, b]*, thenwheref(x) =

*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:

Therefore

which, together with the fact that
* r(x) = 0*
will do the trick.