所屬科目：研究所、轉學考(插大)-高等微積分
(a) , where log means natural logarithm.
(b)
(a) For every x ＞ O, determine Let f(x) be the function given by the pointwise limit you found in part (a).
(b) Fix a positive number [ε. Does converges to f(x) unifornly over [ε, ∞)? Give your reason.
(c) Does converges to f(x) unijormly over (0,∞)? Give your reason.
1An isosceles triangle is a triangle that has t two sides of equal length. 2A metric space M is totally discon nnected if for every there exists a subset U C which is both open and closed, and which is contained in the open ball of radius e centered at x.
(4) [20 分] Let F be a smooth map from R3 to R3. Denote by (x1, x2, x3) the coordinate for the domain R3. Denote the origin (0,0,0) by 0. Suppose that Prove that there exist ●open neighborhoodsof O in (the domain) R3, and a diffeomorphism φ :which maps O to O, ●open neighborhoodsof O in (the target) R3, and a diffeomorphism V which maps O to O, What follows is the diagram for your reference.
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, where log means natural logarithm.
converges to f(x) unifornly over [ε, ∞)? Give your reason.
converges to f(x) unijormly over (0,∞)? Give your reason. 
there exists a subset U C which is both open and closed, and which is contained in the open ball of radius e centered at x.
of O in (the domain) R3, and a diffeomorphism φ :
which maps O to O, ●open neighborhoods
of O in (the target) R3, and a diffeomorphism
V which maps O to O,
