Notes in 03利用等价无穷小求极限

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Published	10/05/2024	一阶等价无穷小1\(x \longrightarrow 0 \text { 时, } \sin x\sim \text{______.}\)
Published	10/05/2024	一阶等价无穷小2\(x \longrightarrow 0 \text { 时, } \tan x \sim \text {_____.}\)
Published	10/05/2024	一阶等价无穷小3\(x \longrightarrow 0 \text { 时, } \arcsin x \sim \text { ______.}\)
Published	10/05/2024	一阶等价无穷小4\(x \longrightarrow 0 \text { 时, } \arctan x \sim \text { ______.}\)
Published	10/05/2024	一阶等价无穷小5\(x \longrightarrow 0 \text { 时, } e^{x}-1 \sim \text{______.}\)
Published	10/05/2024	一阶等价无穷小6\(x \longrightarrow 0 \text { 时, } \ln (1+x) \sim \text { ______. }\)
Published	10/05/2024	一阶等价无穷小7\(x \longrightarrow 0 \text { 时, } \sqrt[n]{1+x}-1 \sim \text{______.}\)
Published	10/05/2024	一阶等价无穷小8\(x \longrightarrow 0 \text { 时, }(1+\beta x)^{\alpha}-1 \sim \text{______.}\)
Published	10/05/2024	一阶等价无穷小9\(x \longrightarrow 0 \text { 时, } \quad a^{x}-1 \sim \text{______.}\)
Published	10/05/2024	一阶等价无穷小10\(x \longrightarrow 0 \text { 时, } \log _{a}(1+x) \sim \text{______.}\)
Published	10/05/2024	二阶等价无穷小1\(x \longrightarrow 0 \text { 时, } 1-\cos x \sim \text { ______.}\)
Published	10/05/2024	二阶等价无穷小2\(x \longrightarrow 0 \text { 时, }\)\(x-\ln (1+x) \sim\text{______.}\)
Published	10/05/2024	三阶等价无穷小1\(x \longrightarrow 0 \text { 时, }\)\(x-\sin x \sim\text{______.}\)
Published	10/05/2024	三阶等价无穷小2\(x \longrightarrow 0 \text { 时, }\)\(\arcsin x-x \sim\text{______.}\)
Published	10/05/2024	三阶等价无穷小3\(x \longrightarrow 0 \text { 时, }\)\(\tan x-x \sim\text{______.}\)
Published	10/05/2024	三阶等价无穷小4\(x \longrightarrow 0 \text { 时, }\)\(x-\arctan x \sim\text{______.}\)
Published	10/05/2024	(1)复合函数等价无穷小\(\text { 当 } x \rightarrow 0 \text { 时, } f(x) \sim a x^{m}, g(x) \sim b x^{n}, a b \neq 0, m, n \text { 为正整数, 则 } f[g(x)] \sim\text{____…
Published	10/05/2024	(2)变上限积分型等价无穷小\(\text { 当 } x \rightarrow 0 \text { 时, } f(x) \sim a x^{m}, a \neq 0, m \text { 为正整数, 则 } \displaystyle\int_{0}^{x} f(t) \mathrm{d} t\…
Published	10/05/2024	(3)复合函数与变上限积分型的等价无穷小\(\text { 当 } x \rightarrow 0 \text { 时, } f(x) \sim a x^{m}, g(x) \sim b x^{n}, a b \neq 0, m, n \text { 为正整数, 则 } \displaystyle\…
Published	10/05/2024	(4)若 \( \displaystyle\lim _{x \rightarrow 0} f(x)=A \neq 0, \displaystyle\lim _{x \rightarrow 0} h(x)=0 \), 且在 \( x \rightarrow 0 \) 时, \( h(x) \neq 0…
Published	10/05/2024	(5)当\( x \rightarrow {x}_{0} \)时,\( \alpha \left( x\right) \sim \beta \left( x\right) \)的充要条件是_____.选自张宇高数18讲
Published	10/05/2024	\(\text { 若 } \alpha \sim \alpha_{1}, \beta \sim \beta_{1}, \text { 则 } \lim \frac{\alpha}{\beta}=\text{______.}\)
Published	10/05/2024	\[\text { 若 } \alpha \sim \alpha_{1}, \beta \sim \beta_{1}, \text { 且 } \lim \frac{\alpha_{1}}{\beta_{1}}=A \neq 1 \text {}\]\[\text{则}\alpha -\beta \…
Published	10/05/2024	序号等价无穷小公式表格1 \(x \longrightarrow 0 \text { 时, }\)\(\sin x \sim \)____2 \(x \longrightarrow 0 \text { 时, }\)\( \tan x \sim \)____3 \(x \…
Published	10/05/2024	请证明: 若 \(\alpha \sim \beta \rightarrow 0^{+}\), 则 \(\ln \alpha \sim \ln \beta \rightarrow-\infty\)(翻译:已知 \(\alpha\) 和 \(\beta\) 是等价无穷小, 则 \(\ln \alpha…
Published	10/05/2024	一阶等价无穷小11\( x \rightarrow 0 \)时,\( \ln \left( {x + \sqrt{1 + {x}^{2}}}\right) \sim \text{______.}\)
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