146 CHAPTER 5. THE DERIVATIVE

x1x2Determine a formula for x2 in terms of the function and its derivative evaluated at x1.The idea is that x2 is a better approximation to a solution to f (x) = 0 than x1. Nowdescribe an iterative procedure which hopefully will yield a sequence of approximatesolutions to f (x)= 0 which converges to a solution to this equation. If you do it right,it is called the Newton Ralphson procedure.

7. Use the above Newton Ralphson procedure to find√

3 valid to four decimal places.

8. Consider the function y= x1/3 which has a zero at x= 0. Show that the above NewtonRalphson method will not work for this example. Is there some condition which willcause the above procedure to work?

9. If x is small and positive, explain why tanx− x > 0. Hint: This amounts to showingthat sinx > xcos(x) . Now use Taylor series approximations.

5.17 L’Hôpital’s RuleThere is an interesting rule which is often useful for evaluating difficult limits. This iscalled L’Hôpital’s3 rule. The best versions of this rule are based on the Cauchy Mean valuetheorem, Theorem 5.11.2 on Page 138.

Theorem 5.17.1 Let [a,b]⊆ [−∞,∞] and suppose f ,g are functions which satisfy,

limx→b−

f (x) = limx→b−

g(x) = 0, (5.8)

and f ′ and g′ exist on (a,b) with g′ (x) ̸= 0 on (a,b). Suppose also that

limx→b−

f ′ (x)g′ (x)

= L. (5.9)

Then

limx→b−

f (x)g(x)

= L. (5.10)

Proof: By the definition of limit and 5.9 there exists c < b such that if t > c, then∣∣∣∣ f ′ (t)g′ (t)

−L∣∣∣∣< ε

2.

3L’Hôpital published the first calculus book in 1696. This rule, named after him, appeared in this book. Therule was actually due to Bernoulli who had been L’Hôpital’s teacher. L’Hôpital did not claim the rule as hisown but Bernoulli accused him of plagiarism. Nevertheless, this rule has become known as L’Hôpital’s rule eversince. There was entirely too much squabbling about who originated various ideas during this period of time. Theversion of the rule presented here is superior to what was discovered by Bernoulli and depends on the Cauchymean value theorem which was found over 100 years after the time of L’Hôpital. Cauchy often saw things whichwere both significant and unobserved by all the others before him. In addition to this, he invented whole new partsof mathematics such as complex analysis and made significant contributions to mechanics and algebra.