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L'Hôpital's Rule by Taylor series Suppose f and g have
Chapter 8, Problem 79AE(choose chapter or problem)
L'Hôpital's Rule by Taylor series Suppose f and g have Taylor series about the point a.
a. If f(a) = g(a) = 0 and \(g^{\prime}(a) \neq 0\), evaluate \(\lim \limits_{x \rightarrow a} f(x) / g(x)\) by expanding f and g in their Taylor series. Show that the result is consistent with l'Hôpital's Rule.
b. If \(f(a)=g(a)=f^{\prime}(a)=g^{\prime}(a)=0\) and \(g^{\prime \prime}(a) \neq 0\), evaluate \(\lim \limits_{x \rightarrow a} \frac{f(x)}{g(x)}\) by expanding f and g in their Taylor series. Show that the result is consistent with two applications of l'Hôpital's Rule.
Questions & Answers
QUESTION:
L'Hôpital's Rule by Taylor series Suppose f and g have Taylor series about the point a.
a. If f(a) = g(a) = 0 and \(g^{\prime}(a) \neq 0\), evaluate \(\lim \limits_{x \rightarrow a} f(x) / g(x)\) by expanding f and g in their Taylor series. Show that the result is consistent with l'Hôpital's Rule.
b. If \(f(a)=g(a)=f^{\prime}(a)=g^{\prime}(a)=0\) and \(g^{\prime \prime}(a) \neq 0\), evaluate \(\lim \limits_{x \rightarrow a} \frac{f(x)}{g(x)}\) by expanding f and g in their Taylor series. Show that the result is consistent with two applications of l'Hôpital's Rule.
ANSWER:Solu