1.lim x->∞ （2x/x^2+1)*cosx 2..lim x->∞{ [(x^2-x)arctanx]/x^3-x-5}3..lim x->0 （e^x-e^-x-2x)/(x-sinx) 4..lim x->0 (sinx-x)/(x^3) 5..lim x->0 (x^2*tanx)/(x-tanx) 6..lim x->0 (1+3/x)^2x 7..lim x->0 (1+2x)^(3/sinx) 8.lim x->∞ (x/1+x)^(x-3) 9..li

1.lim x->∞ （2x/x^2+1)*cosx 2..lim x->∞{ [(x^2-x)arctanx]/x^3-x-5}3..lim x->0 （e^x-e^-x-2x)/(x-sinx) 4..lim x->0 (sinx-x)/(x^3) 5..lim x->0 (x^2*tanx)/(x-tanx) 6..lim x->0 (1+3/x)^2x 7..lim x->0 (1+2x)^(3/sinx) 8.lim x->∞ (x/1+x)^(x-3) 9..li
1.lim x->∞ （2x/x^2+1)*cosx 2..lim x->∞{ [(x^2-x)arctanx]/x^3-x-5}
3..lim x->0 （e^x-e^-x-2x)/(x-sinx) 4..lim x->0 (sinx-x)/(x^3) 5..lim x->0 (x^2*tanx)/(x-tanx) 6..lim x->0 (1+3/x)^2x 7..lim x->0 (1+2x)^(3/sinx) 8.lim x->∞ (x/1+x)^(x-3) 9..lim x->0 (1/(1-x)-3/(1-x^3)) 10..lim x->1 (1/(x-1)-1/lnx)

1.lim x->∞ （2x/x^2+1)*cosx 2..lim x->∞{ [(x^2-x)arctanx]/x^3-x-5}3..lim x->0 （e^x-e^-x-2x)/(x-sinx) 4..lim x->0 (sinx-x)/(x^3) 5..lim x->0 (x^2*tanx)/(x-tanx) 6..lim x->0 (1+3/x)^2x 7..lim x->0 (1+2x)^(3/sinx) 8.lim x->∞ (x/1+x)^(x-3) 9..li
1.
lim(x→∞) 2x/(x² + 1) * cosx
= lim(x→∞) 2/(x + 1/x) * cosx
1/x → 0、cosx不定式、2/(x + 1/x) → 2/∞ → 0
原式 = 0
2.
lim(x→∞) [(x² - x)arctanx]/(x³ - x - 5)、∞/∞
= lim(x→∞) [(x² - x) * π/2]/(x³ - x - 5)、当x→∞时arctanx → π/2
= lim(x→∞) [(1 - 1/x) * π/2]/(x - 1/x - 5/x²)、上下除以x²
= [(1 - 0) * π/2]/(∞ - 0 - 0)
= 0
3.
lim(x→0) [e^x - e^(- x) - 2x]/(x - sinx)、0/0
= lim(x→0) [e^x + e^(- x) - 2]/(1 - cosx)、0/0
= lim(x→0) [e^x - e^(- x)]/sinx、0/0
= lim(x→0) [e^x + e^(- x)]/cosx
= (1 + 1)/1
= 2
4.
lim(x→0) (sinx - x)/x³、0/0
= lim(x→0) (cosx - 1)/(3x²)、0/0
= lim(x→0) (- sinx)/(6x)
= (- 1/6)lim(x→0) (sinx)/x
= - 1/6
5.
lim(x→0) (x²tanx)/(x - tanx)
= lim(x→0) (x² * sinx/cosx)/(x - sinx/cosx)
= lim(x→0) (sinx)/x * x³/(xcosx - sinx)
= lim(x→0) x³/(xcosx - sinx)、0/0
= lim(x→0) 3x²/(cosx - xsinx - cosx)
= lim(x→0) 3x²/(- xsinx)
= lim(x→0) (- 3) * x/sinx
= - 3
6.
lim(x→0) (1 + 3/x)^(2x)
= lim(x→0) (1 + 3/x)^(x/3 * 3/x * 2x)
= lim(x→0) 1^(6)、极限lim(x→0) (1 + 1/x)^x = 1
= 1
7.
lim(x→0) (1 + 2x)^(3/sinx)
= lim(x→0) (1 + 2x)^[1/(2x) * 2x * 3/sinx]
= e^lim(x→0) (6 * x/sinx)
= e⁶
8.
lim(x→∞) [x/(1 + x)]^(x - 3)
= lim(x→∞) [(x + 1 - 1)/(1 + x)]^(x - 3)
= lim(x→∞) [1 + 1/(- (1 + x))]^(x - 3)
= lim(x→∞) [1 + 1/(- (1 + x))]^[- (1 + x) * - 1/(1 + x) * (x - 3)]
= e^lim(x→∞) (3 - x)/(x + 1)
= e^lim(x→∞) (3/x - 1)/(1 + 1/x)
= e^(- 1)
= 1/e
9.
lim(x→0) [1/(1 - x) - 3/(1 - x³)]
= lim(x→0) {1/(1 - x) - 3/[(1 - x)(1 + x + x²)]}
= lim(x→0) [(1 + x + x²) - 3]/[(1 - x)(1 + x + x²)]
= lim(x→0) (x² + x - 2)/[(1 - x)(1 + x + x²)]
= lim(x→0) [(x - 1)(x + 2)]/[(1 - x)(1 + x + x²)]
= lim(x→0) - (x + 2)/(1 + x + x²)
= - (0 + 2)/(1 + 0 + 0)
= - 2
10.
lim(x→1) [1/(x - 1) - 1/lnx]
= lim(x→1) [lnx - (x - 1)]/[(x - 1)lnx]、0/0
= lim(x→1) (1/x - 1)/(- 1/x + lnx + 1)、0/0
= lim(x→1) (- 1/x²)/(1/x² + 1/x)
= lim(x→1) - 1/(1 + x)
= - 1/(1 + 1)
= - 1/2
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