求极限lim arctan(x^2-2x)/sin3πx.当x趋向2时,

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求极限limarctan(x^2-2x)/sin3πx.当x趋向2时,求极限limarctan(x^2-2x)/sin3πx.当x趋向2时,求极限limarctan(x^2-2x)/sin3πx.当x

求极限lim arctan(x^2-2x)/sin3πx.当x趋向2时,
求极限lim arctan(x^2-2x)/sin3πx.
当x趋向2时,

求极限lim arctan(x^2-2x)/sin3πx.当x趋向2时,
sin3πx=sin(3πx-6π),
当x->2, 3πx-6π->0, 所以3πx-6π是sin3πx=sin(3πx-6π)的等价无穷小,可以用等价无穷小的代换.
lim arctan(x^2-2x)/sin3πx
=lim(x^2-2x)/(3πx-6π)
=limx(x-2)/[3π(x-2)]
=lim[x/(3π)]
=2/(3π)

lim(x->2) arctan(x^2-2x)/sin(3πx) (0/0)
=lim(x->2) {(2x-2)/[1+(x^2-2x)^2]} / [3πcos(3πx)]
=2/(3π)