# 2. Continuity, Differentiability, Series, and Scaling Consider the...

## Transcribed Text

2. Continuity, Differentiability, Series, and Scaling Consider the function f(x) : R R,f(x= cos(sin(bxalog()x())),with b a constant, O > 0, and log(-) the natural logarithm. (a) Is f(x) continuous at x = 0? State any restrictions on a, b necessary for this property to hold. (b) Is f(x) differentiable at . = 0? State any restrictions on a, b necessary for this property to hold. (c) Use some combination of the rules for differentiation to obtain the first two terms of a series expansion for f(x) valid in the neighborhood of x = 0. Is this a Taylor series? Is it a Taylor-Laurent series? Hint: treat the argument of sin(-) as a variable y(x). (d) Use the "big O" o(.) notation to represent the next term in the series in (c). The argument of o(.) should incorporate (powers of) b, x, log(x), etc. so that the only refinement necessary for you to obtain a fully valid three-term series expansion of f(x) is a fixed, constant coefficient independent of the parameters of this problem.

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