作业帮高数极限和定积分问题~急~x→0时,求limex^2-1-x^2/x^2(ex^2-1)∫sin3xsin5xdx=?X→∞时,求lim（2/∏-arctantx)^x第3个问题应该是lim[(2arctanx)/∏]^x
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![高数极限和定积分问题~急~x→0时,求limex^2-1-x^2/x^2(ex^2-1)∫sin3xsin5xdx=?X→∞时,求lim（2/∏-arctantx)^x第3个问题应该是lim[(2arctanx)/∏]^x](/uploads/image/z/1825602-42-2.jpg?t=%E9%AB%98%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E5%AE%9A%E7%A7%AF%E5%88%86%E9%97%AE%E9%A2%98%7E%E6%80%A5%7Ex%E2%86%920%E6%97%B6%2C%E6%B1%82limex%5E2-1-x%5E2%2Fx%5E2%28ex%5E2-1%29%E2%88%ABsin3xsin5xdx%3D%3FX%E2%86%92%E2%88%9E%E6%97%B6%2C%E6%B1%82lim%EF%BC%882%2F%E2%88%8F-arctantx%29%5Ex%E7%AC%AC3%E4%B8%AA%E9%97%AE%E9%A2%98%E5%BA%94%E8%AF%A5%E6%98%AFlim%5B%282arctanx%29%2F%E2%88%8F%5D%5Ex)
高数极限和定积分问题~急~x→0时,求limex^2-1-x^2/x^2(ex^2-1)∫sin3xsin5xdx=?X→∞时,求lim（2/∏-arctantx)^x第3个问题应该是lim[(2arctanx)/∏]^x
高数极限和定积分问题~急~
x→0时,求limex^2-1-x^2/x^2(ex^2-1)
∫sin3xsin5xdx=?
X→∞时,求lim（2/∏-arctantx)^x
第3个问题应该是lim[(2arctanx)/∏]^x
高数极限和定积分问题~急~x→0时,求limex^2-1-x^2/x^2(ex^2-1)∫sin3xsin5xdx=?X→∞时,求lim（2/∏-arctantx)^x第3个问题应该是lim[(2arctanx)/∏]^x
罗比塔法则：
上下各自求导
lim(2xex^2-2x)/(2x(ex^2-1)+x^2ex^2)
=lim(2ex^2-2)/(2(ex^2-1)+xex^2)
继续求导：
=lim4xex^2/(4xex^2+2x^2ex^2)
=lim4ex^2/(4ex^2+2xex^2)
=4/(4+0)
=1
积化和差
∫sin3xsin5xdx=
∫1/2(cos2x-cos8x)dx=
1/4sin2x-1/16sin8x+c
lim（2arctanx/π)^x
=lim（1+(2arctanx-π/π))^x
=lim[（1+(2arctanx-π/π))^[1/(2arctanx-π/π))](x(2arctanx-π/π))
=e^(x(2arctanx-π/π))
limx(2arctanx-π/π)
=lim(2arctanx-π/π)/(1/x)
罗比塔法则：
上下各自求导
=lim(2/π(1+x^2))/(-1/x^2)
=lim(-2x^2/π(1+x^2))
=-2/π
所以原式=e^(-2/π)
1.
lim （e^x^2-1-x^2)/x^2(e^x^2-1)
=lim （e^x^2-1-x^2)/x^4 (e^x^2-1等价无穷小为x^2)
=lim 2x(e^x^2-1)/4x^3 (洛必达法则)
=lim x*x^2/2x^3 (e^x^2-1等价无穷小为x^2)
=1/2
2...
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1.
lim （e^x^2-1-x^2)/x^2(e^x^2-1)
=lim （e^x^2-1-x^2)/x^4 (e^x^2-1等价无穷小为x^2)
=lim 2x(e^x^2-1)/4x^3 (洛必达法则)
=lim x*x^2/2x^3 (e^x^2-1等价无穷小为x^2)
=1/2
2.
sin3xsin5x=[cos(5x-3x)-cos(5x+3x)]/2=[cos2x-cos8x]/2
∫sin3xsin5xdx=1/2∫[cos2x-cos8x]dx
=(1/2)*[sin2x/2-sin8x/8]
=sin2x/4-sin8x/16
3.
0<2/∏-arctantx<1,指数又趋于正无穷
∴lim（2/∏-arctantx)^x =0
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