【1991-12-3 分】 已知当 x → 0 \displaystyle x \to 0 x → 0 ( 1 + a x 2 ) 1 3 − 1 \displaystyle (1+a x^{2})^{\frac{1}{3}}-1 ( 1 + a x 2 ) 3 1 − 1 cos x − 1 \displaystyle \cos x-1 cos x − 1 a = \displaystyle a= a =
【1992-5-3 分】 当 x → 0 \displaystyle x \to 0 x → 0 x 2 \displaystyle x^{2} x 2 1 − cos x \displaystyle 1 - \cos x 1 − cos x 1 − x 2 − 1 \displaystyle \sqrt{1-x^{2}}-1 1 − x 2 − 1 x − sin x \displaystyle x-\sin x x − sin x
【1992-4-3 分】 当 x → 0 \displaystyle x \to 0 x → 0 x 2 \displaystyle x^{2} x 2 1 − cos x \displaystyle 1 - \cos x 1 − cos x 1 − x 2 − 1 \displaystyle \sqrt{1-x^{2}}-1 1 − x 2 − 1 x − tan x \displaystyle x-\tan x x − tan x
【1992-3-3 分】 当 x → 0 \displaystyle x \to 0 x → 0 x − sin x \displaystyle x-\sin x x − sin x x 2 \displaystyle x^{2} x 2
【1993-12-3 分】 设 f ( x ) = ∫ 0 sin x sin ( t 2 ) d t , g ( x ) = x 3 + x 4 \displaystyle f(x)=\int_{0}^{\sin x} \sin (t^{2}) d t,g(x)=x^{3}+x^{4} f ( x ) = ∫ 0 s i n x sin ( t 2 ) d t , g ( x ) = x 3 + x 4 x → 0 \displaystyle x \to 0 x → 0 f ( x ) \displaystyle f(x) f ( x ) g ( x ) \displaystyle g(x) g ( x )
【1996-3-3 分】 设当 x → 0 \displaystyle x \to 0 x → 0 e x − ( a x 2 + b x + 1 ) \displaystyle e^{x}-(a x^{2}+b x+1) e x − ( a x 2 + b x + 1 ) x 2 \displaystyle x^{2} x 2 a = 1 2 , b = 1 \displaystyle a=\dfrac{1}{2}, b=1 a = 2 1 , b = 1 a = 1 , b = 1 \displaystyle a=1, b=1 a = 1 , b = 1 a = − 1 2 , b = − 1 \displaystyle a=-\dfrac{1}{2}, b=-1 a = − 2 1 , b = − 1 a = − 1 , b = 1 \displaystyle a=-1, b=1 a = − 1 , b = 1
【1996-12-3 分】 设 f ( x ) \displaystyle f(x) f ( x ) f ( 0 ) = 0 \displaystyle f(0)=0 f ( 0 ) = 0 f ′ ( 0 ) ≠ 0 \displaystyle f'(0) \ne0 f ′ ( 0 ) = 0 F ( x ) = ∫ 0 x ( x 2 − t 2 ) f ( t ) d t \displaystyle F(x)=\int_{0}^{x}(x^{2}-t^{2}) f(t) d t F ( x ) = ∫ 0 x ( x 2 − t 2 ) f ( t ) d t x → 0 \displaystyle x \to 0 x → 0 F ′ ( x ) \displaystyle F'(x) F ′ ( x ) x k \displaystyle x^{k} x k
【1997-4-3 分】 设 f ( x ) \displaystyle f(x) f ( x ) φ ( x ) \displaystyle \varphi(x) φ ( x ) x = 0 \displaystyle x=0 x = 0 x → 0 \displaystyle x \to 0 x → 0 ∫ 0 x f ( t ) sin t d t \displaystyle \int_{0}^{x} f(t) \sin t d t ∫ 0 x f ( t ) sin t d t ∫ 0 x t φ ( t ) d t \displaystyle \int_{0}^{x} t \varphi(t) d t ∫ 0 x tφ ( t ) d t x → 0 \displaystyle x \to 0 x → 0 f ( x ) \displaystyle f(x) f ( x ) φ ( x ) \displaystyle \varphi(x) φ ( x )
【1997-3-3 分】 设 f ( x ) = ∫ 0 1 − cos x sin t 2 d t , g ( x ) = x 5 5 + x 6 6 \displaystyle f(x)=\int_{0}^{1-\cos x} \sin t^{2} d t,g(x)=\dfrac{x^{5}}{5}+\dfrac{x^{6}}{6} f ( x ) = ∫ 0 1 − c o s x sin t 2 d t , g ( x ) = 5 x 5 + 6 x 6 x → 0 \displaystyle x \to 0 x → 0 f ( x ) \displaystyle f(x) f ( x ) g ( x ) \displaystyle g(x) g ( x )
【1997-2-3 分】 设当 x → 0 \displaystyle x \to 0 x → 0 e tan x − e x \displaystyle e^{\tan x}-e^{x} e t a n x − e x x n \displaystyle x^{n} x n
【1999-2-3 分】 设 α ( x ) = ∫ 0 5 x sin t t d t , β ( x ) = ∫ 0 sin x ( 1 + t ) 1 t d t \displaystyle \alpha(x)=\int_{0}^{5 x} \dfrac{\sin t}{t} d t,\beta(x)=\int_{0}^{\sin x}(1+t)^{\frac{1}{t}} d t α ( x ) = ∫ 0 5 x t sin t d t , β ( x ) = ∫ 0 s i n x ( 1 + t ) t 1 d t x → 0 \displaystyle x \to 0 x → 0 α ( x ) \displaystyle \alpha(x) α ( x ) β ( x ) \displaystyle \beta(x) β ( x )
【2001-2-3 分】 设当 x → 0 \displaystyle x \to 0 x → 0 ( 1 − cos x ) ln ( 1 + x 2 ) \displaystyle (1-\cos x) \ln (1+x^{2}) ( 1 − cos x ) ln ( 1 + x 2 ) x sin x n \displaystyle x \sin x^{n} x sin x n x sin x n \displaystyle x \sin x^{n} x sin x n ( e x 2 − 1 ) \displaystyle (e^{x^{2}}-1) ( e x 2 − 1 )
【2003-2-4 分】 若 x → 0 \displaystyle x \to 0 x → 0 ( 1 − a x 2 ) 1 4 − 1 \displaystyle (1-a x^{2})^{\frac{1}{4}}-1 ( 1 − a x 2 ) 4 1 − 1 x sin x \displaystyle x \sin x x sin x a = \displaystyle a= a =
【2004-12-4 分】 把 x → 0 + \displaystyle x \to 0^{+} x → 0 + α = ∫ 0 x cos t 2 d t , β = ∫ 0 x 2 tan t d t , γ = ∫ 0 x sin t 3 d t \displaystyle \alpha=\int_{0}^{x} \cos t^{2} d t,\beta=\int_{0}^{x^{2}} \tan \sqrt{t} d t,\gamma=\int_{0}^{\sqrt{x}} \sin t^{3} d t α = ∫ 0 x cos t 2 d t , β = ∫ 0 x 2 tan t d t , γ = ∫ 0 x sin t 3 d t α , β , γ \displaystyle \alpha, \beta, \gamma α , β , γ α , γ , β \displaystyle \alpha, \gamma, \beta α , γ , β β , α , γ \displaystyle \beta, \alpha, \gamma β , α , γ β , γ , α \displaystyle \beta, \gamma, \alpha β , γ , α
【2005-2-4 分】 当 x → 0 \displaystyle x \to 0 x → 0 α ( x ) = k x 2 \displaystyle \alpha(x)=k x^{2} α ( x ) = k x 2 β ( x ) = 1 + x arcsin x − cos x \displaystyle \beta(x)=\sqrt{1+x \arcsin x}-\sqrt{\cos x} β ( x ) = 1 + x arcsin x − cos x k = \displaystyle k= k =
【2007-1234-4 分】 当 x → 0 + \displaystyle x \to 0^{+} x → 0 + x \displaystyle \sqrt{x} x 1 − e x \displaystyle 1-e^{\sqrt{x}} 1 − e x ln 1 + x 1 − x \displaystyle \ln \dfrac{1+x}{1-\sqrt{x}} ln 1 − x 1 + x 1 + x − 1 \displaystyle \sqrt{1+\sqrt{x}}-1 1 + x − 1 1 − cos x \displaystyle 1-\cos \sqrt{x} 1 − cos x
【2009-123-4 分】 当 x → 0 \displaystyle x \to 0 x → 0 f ( x ) = x − sin a x \displaystyle f(x)=x-\sin a x f ( x ) = x − sin a x g ( x ) = x 2 ln ( 1 − b x ) \displaystyle g(x)=x^{2} \ln (1-b x) g ( x ) = x 2 ln ( 1 − b x ) a = 1 , b = − 1 6 \displaystyle a=1, b=-\dfrac{1}{6} a = 1 , b = − 6 1 a = 1 , b = 1 6 \displaystyle a=1, b=\dfrac{1}{6} a = 1 , b = 6 1 a = − 1 , b = − 1 6 \displaystyle a=-1, b=-\dfrac{1}{6} a = − 1 , b = − 6 1 a = − 1 , b = 1 6 \displaystyle a=-1, b=\dfrac{1}{6} a = − 1 , b = 6 1
【2011-23-4 分】 已知当 x → 0 \displaystyle x \to 0 x → 0 f ( x ) = 3 sin x − sin 3 x \displaystyle f(x)=3 \sin x-\sin 3 x f ( x ) = 3 sin x − sin 3 x c x k \displaystyle c x^{k} c x k k = 1 , c = 4 \displaystyle k=1, c=4 k = 1 , c = 4 k = 1 , c = − 4 \displaystyle k=1, c=-4 k = 1 , c = − 4 k = 3 , c = 4 \displaystyle k=3, c=4 k = 3 , c = 4 k = 3 , c = − 4 \displaystyle k=3, c=-4 k = 3 , c = − 4
【2013-2-4 分】 设 cos x − 1 = x sin α ( x ) \displaystyle \cos x-1=x \sin \alpha(x) cos x − 1 = x sin α ( x ) ∣ α ( x ) ∣ < π 2 \displaystyle |\alpha(x)|<\dfrac{\pi}{2} ∣ α ( x ) ∣ < 2 π x → 0 \displaystyle x \to 0 x → 0 α ( x ) \displaystyle \alpha(x) α ( x )
【2013-3-4 分】 当 x → 0 \displaystyle x \to 0 x → 0 o ( x ) \displaystyle o(x) o ( x ) x \displaystyle x x x ⋅ o ( x 2 ) = o ( x 3 ) \displaystyle x \cdot o\left(x^{2}\right)=o\left(x^{3}\right) x ⋅ o ( x 2 ) = o ( x 3 ) o ( x ) ⋅ o ( x 2 ) = o ( x 3 ) \displaystyle o(x) \cdot o\left(x^{2}\right)=o\left(x^{3}\right) o ( x ) ⋅ o ( x 2 ) = o ( x 3 ) o ( x 2 ) + o ( x 2 ) = o ( x 2 ) \displaystyle o\left(x^{2}\right)+o\left(x^{2}\right)=o\left(x^{2}\right) o ( x 2 ) + o ( x 2 ) = o ( x 2 ) o ( x ) + o ( x 2 ) = o ( x 2 ) \displaystyle o(x)+o\left(x^{2}\right)=o\left(x^{2}\right) o ( x ) + o ( x 2 ) = o ( x 2 )
【2014-2-4 分】 当 x → 0 + \displaystyle x \to 0^{+} x → 0 + ln α ( 1 + 2 x ) \displaystyle \ln^{\alpha}(1+2x) ln α ( 1 + 2 x ) ( 1 − cos x ) 1 α \displaystyle (1-\cos x)^{\frac{1}{\alpha}} ( 1 − cos x ) α 1 ( 2 , + ∞ ) \displaystyle (2,+\infty) ( 2 , + ∞ ) ( 1 , 2 ) \displaystyle (1,2) ( 1 , 2 ) ( 1 2 , 1 ) \displaystyle \left(\dfrac{1}{2}, 1\right) ( 2 1 , 1 ) ( 0 , 1 2 ) \displaystyle \left(0, \dfrac{1}{2}\right) ( 0 , 2 1 )
【2014-3-4 分】 设 p ( x ) = a + b x + c x 2 + d x 3 \displaystyle p(x)=a+b x+c x^{2}+dx^{3} p ( x ) = a + b x + c x 2 + d x 3 x → 0 \displaystyle x \to 0 x → 0 p ( x ) − tan x \displaystyle p(x)-\tan x p ( x ) − tan x x 3 \displaystyle x^{3} x 3 a = 0 \displaystyle a=0 a = 0 b = 1 \displaystyle b=1 b = 1 c = 0 \displaystyle c=0 c = 0 d = 1 6 \displaystyle d=\dfrac{1}{6} d = 6 1
【2016-2-4 分】 设 α 1 = x ( cos x − 1 ) \displaystyle \alpha_{1}=x(\cos \sqrt{x}-1) α 1 = x ( cos x − 1 ) α 2 = x ln ( 1 + x 3 ) \displaystyle \alpha_{2}=\sqrt{x} \ln (1+\sqrt[3]{x}) α 2 = x ln ( 1 + 3 x ) α 3 = x + 1 3 − 1 \displaystyle \alpha_{3}=\sqrt[3]{x+1}-1 α 3 = 3 x + 1 − 1 x → 0 + \displaystyle x \to 0^{+} x → 0 + α 1 , α 2 , α 3 \displaystyle \alpha_{1}, \alpha_{2}, \alpha_{3} α 1 , α 2 , α 3 α 2 , α 3 , α 1 \displaystyle \alpha_{2}, \alpha_{3}, \alpha_{1} α 2 , α 3 , α 1 α 2 , α 1 , α 3 \displaystyle \alpha_{2}, \alpha_{1}, \alpha_{3} α 2 , α 1 , α 3 α 3 , α 2 , α 1 \displaystyle \alpha_{3}, \alpha_{2}, \alpha_{1} α 3 , α 2 , α 1
【2019-123-4 分】 当 x → 0 \displaystyle x \to 0 x → 0 x − tan x \displaystyle x-\tan x x − tan x x k \displaystyle x^{k} x k k = ( ) \displaystyle k=( ) k = ( )
【2020-12-4 分】 当 x → 0 + \displaystyle x \to 0^{+} x → 0 + ∫ 0 x ( e t 2 − 1 ) d t \displaystyle \int_{0}^{x}\left(e^{t^{2}}-1\right) d t ∫ 0 x ( e t 2 − 1 ) d t ∫ 0 x ln ( 1 + t 3 ) d t \displaystyle \int_{0}^{x} \ln \left(1+\sqrt{t^{3}}\right) d t ∫ 0 x ln ( 1 + t 3 ) d t ∫ 0 sin x sin t 2 d t \displaystyle \int_{0}^{\sin x} \sin t^{2} d t ∫ 0 s i n x sin t 2 d t ∫ 0 1 − cos x sin 3 t d t \displaystyle \int_{0}^{1-\cos x} \sqrt{\sin^{3} t} d t ∫ 0 1 − c o s x sin 3 t d t
【2021-23-5 分】 当 x → 0 \displaystyle x \to 0 x → 0 ∫ 0 x 2 ( e t 3 − 1 ) d t \displaystyle \int_{0}^{x^{2}}(e^{t^{3}}-1) d t ∫ 0 x 2 ( e t 3 − 1 ) d t x 7 \displaystyle x^{7} x 7
【2021-1-5 分】 设函数 f ( x ) = sin x 1 + x 2 \displaystyle f(x)=\dfrac{\sin x}{1+x^{2}} f ( x ) = 1 + x 2 sin x x = 0 \displaystyle x=0 x = 0 a x + b x 2 + c x 3 \displaystyle ax+b x^{2}+c x^{3} a x + b x 2 + c x 3 a = 1 , b = 0 , c = − 7 6 \displaystyle a=1, b=0, c=-\dfrac{7}{6} a = 1 , b = 0 , c = − 6 7 a = 1 , b = 0 , c = 7 6 \displaystyle a=1, b=0, c=\dfrac{7}{6} a = 1 , b = 0 , c = 6 7 a = − 1 , b = − 1 , c = − 7 6 \displaystyle a=-1, b=-1, c=-\dfrac{7}{6} a = − 1 , b = − 1 , c = − 6 7 a = − 1 , b = − 1 , c = 7 6 \displaystyle a=-1, b=-1, c=\dfrac{7}{6} a = − 1 , b = − 1 , c = 6 7
【2022-23-5 分】 当 x → 0 \displaystyle x \to 0 x → 0 α ( x ) \displaystyle \alpha(x) α ( x ) β ( x ) \displaystyle \beta(x) β ( x ) α ( x ) β ( x ) \displaystyle \alpha(x) ~ \beta(x) α ( x ) β ( x ) α 2 ( x ) β 2 ( x ) \displaystyle \alpha^{2}(x) ~ \beta^{2}(x) α 2 ( x ) β 2 ( x ) α 2 ( x ) β 2 ( x ) \displaystyle \alpha^{2}(x) ~ \beta^{2}(x) α 2 ( x ) β 2 ( x ) α ( x ) β ( x ) \displaystyle \alpha(x) ~ \beta(x) α ( x ) β ( x ) α ( x ) β ( x ) \displaystyle \alpha(x) ~ \beta(x) α ( x ) β ( x ) α ( x ) − β ( x ) = o ( α ( x ) ) \displaystyle \alpha(x)-\beta(x)=o(\alpha(x)) α ( x ) − β ( x ) = o ( α ( x )) α ( x ) − β ( x ) = o ( α ( x ) ) \displaystyle \alpha(x)-\beta(x)=o(\alpha(x)) α ( x ) − β ( x ) = o ( α ( x )) α ( x ) β ( x ) \displaystyle \alpha(x) ~ \beta(x) α ( x ) β ( x )
【2023-12-5 分】 当 x → 0 \displaystyle x \to 0 x → 0 f ( x ) = a x + b x 2 + ln ( 1 + x ) \displaystyle f(x)=a x+b x^{2}+\ln (1+x) f ( x ) = a x + b x 2 + ln ( 1 + x ) g ( x ) = e x 2 − cos x \displaystyle g(x)=e^{x^{2}}-\cos x g ( x ) = e x 2 − cos x a b = \displaystyle a b= ab =
【2024-3-5 分】 当 x → 0 \displaystyle x \to 0 x → 0 ∫ 0 x ( 1 + t 2 ) sin t 2 1 + cos t 2 d t \displaystyle \int_{0}^{x} \dfrac{(1+t^{2}) \sin t^{2}}{1+\cos t^{2}} d t ∫ 0 x 1 + cos t 2 ( 1 + t 2 ) sin t 2 d t x k \displaystyle x^{k} x k k = \displaystyle k= k =
【2025-3-5 分】 当 x → 0 + \displaystyle x \to 0^{+} x → 0 + e − sin x − 1 \displaystyle e^{-\sin x}-1 e − s i n x − 1 x + 1 − cos x \displaystyle \sqrt{x+1}-\cos x x + 1 − cos x 1 − cos 2 x \displaystyle 1-\cos \sqrt{2 x} 1 − cos 2 x 1 − ln ( 1 + x ) x \displaystyle 1-\dfrac{\ln (1+x)}{x} 1 − x ln ( 1 + x )
【2025-2-5 分】 设函数 f ( x ) \displaystyle f(x) f ( x ) g ( x ) \displaystyle g(x) g ( x ) x = 0 \displaystyle x=0 x = 0 x → 0 \displaystyle x \to 0 x → 0 f ( x ) \displaystyle f(x) f ( x ) g ( x ) \displaystyle g(x) g ( x ) x → 0 \displaystyle x \to 0 x → 0 f ( x ) + g ( x ) = o ( g ( x ) ) \displaystyle f(x)+g(x)=o(g(x)) f ( x ) + g ( x ) = o ( g ( x )) f ( x ) g ( x ) = o ( f 2 ( x ) ) \displaystyle f(x) g(x)=o(f^{2}(x)) f ( x ) g ( x ) = o ( f 2 ( x )) f ( x ) = o ( e g ( x ) − 1 ) \displaystyle f(x)=o(e^{g(x)}-1) f ( x ) = o ( e g ( x ) − 1 ) f ( x ) = o ( g 2 ( x ) ) \displaystyle f(x)=o(g^{2}(x)) f ( x ) = o ( g 2 ( x ))