高等数学等价无穷小替换时,sinx~x,那么(sinx)2可以替换为x2(平方)吗? 高等数学等价无穷小问题。 sinx等价于x 那么(sinx)...
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sinx-tanx=tanx(cosx-1)~x*(-x^2/2)=-x^3/2(x->0)
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4\u30011-cos(x^2)\uff5e1/2x^4 (x\u21920)
5\u3001sinx~x (x\u21920)
6\u3001tanx~x (x\u21920)
7\u3001arcsinx~x (x\u21920)
8\u3001arctanx~x (x\u21920)
9\u30011-cosx~1/2x^2 (x\u21920)
高等数学等价无穷小替换时,sinx~x,那么(sinx)2可以替换为x2(平方)。
当x→0时,sinx的泰勒展开式为sinx=x+o(x)
o(x)指的是x的高阶无穷小,所以当x→0时
可以(sinx)~x当x→0时(sinx)²=x²+o(x²)
所以当x→0时,可以(sinx)²~x²。
例题:
limx→0(sinx-tanx)/{[3√(1+X^2)-1][(1+sinx)-1]}
分母部分可以用等价无穷小替换为“X^2/3"和”sinx/3
分母替换是正确的,sinx/3可继续替换为x/3.分子这样做:
sinx-tanx=tanx(cosx-1)~x*(-x^2/2)=-x^3/2(x->0)
所以最终答案为lim{x->0}(-x^3/2)/(x^3/9)=-9/2.
x→0)sinx+(sinx)^2→01+sinx→(1+sinx)^2(1+sinx)^(1/2)-1→1+sinx-1→sinx
x无穷小时,1+sinx和1+2sinx+(sinx)^2非常接近。
其差量sinx+(sinx)^2无穷小,因此用1+2sinx+(sinx)^2代替1+sinx,平方根(1+sinx)-1,得sinx。
扩展资料
高等数学中所有等价无穷小的公式:
当x→0,且x≠0,则
x~sinx~tanx~arcsinx~arctanx;
x~ln(1+x)~(e^x-1);
(1-cosx)~x*x/2;
[(1+x)^n-1]~nx;
loga(1+x)~x/lna;
a的x次方~xlna;
(1+x)的1/n次方~1/nx(n为正整数);
注:^是乘方,~是等价于,这是我做题的时候总结出来的.
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