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Multiplicity of a Root and Taylor's Theorem

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Let f(x) be defined as

f(x) = tanx/x if x =/ 0
= 1 if x = 0

and let g(x) = f(x) - 1. Then g(x) is continuous at x = 0 and, in fact, g(x) has derivatives of all orders at x = 0. Determine the multiplicity of the root g(x) has at x = 0. Hint: Apply Taylor's Theorem.

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This solution looks at how to express tan(x)/x as a taylor series about x=0 using Lhopital rule; then you factor it out of the sum and you will see the multiplicity. This solution is provided in an attached PDF and Word copy.

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express tan(x)/x as a taylor series about x=0. This is a ...

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