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大一微积分极限题,求解
2个回答
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(1)
lim(x->0) [√(x^2+1) -1] /[√(x^2+9) -3]
=lim(x->0) [(x^2+1) -1] /{ [√(x^2+9) -3]. [√(x^2+1) +1] }
=(1/2)lim(x->0) x^2 / [√(x^2+9) -3]
=(1/2)lim(x->0) x^2 .[√(x^2+9) +3]/ [(x^2+9) -9]
=(1/2)lim(x->0) [√(x^2+9) +3]
=6/2
=3
(2)
x^(1/3) -1 = [ x^(1/6)-1 ] [ x^(1/6) +1 ]
[ x^(1/2) -1 ] = [ x^(1/6) -1 ] [ x^(1/3) + x^(1/6) +1 ]
lim(x->1) [ x^(1/3) -1 ]/ [ x^(1/2) -1 ]
=lim(x->1) [ x^(1/6)-1 ] [ x^(1/6) +1 ]/ { [ x^(1/6) -1 ] [ x^(1/3) + x^(1/6) +1 ] }
=lim(x->1) [ x^(1/6) +1 ]/ [ x^(1/3) + x^(1/6) +1 ]
=2/3
(3)
lim(x->∞) { √[(x+2)(x-1)] -x }
=lim(x->∞) [(x+2)(x-1) -x^2 ]/ { √[(x+2)(x-1)] +x }
=lim(x->∞) (x-2)/ { √[(x+2)(x-1)] +x }
=lim(x->∞) (1- 2/x)/ { √[(1+2/x)(1-1/x)] +1 }
= 1/( 1+1)
=1/2
(4)
lim(x->∞) (2x-1)^30.(3x-1)^20/ (2x+1)^50
分子分母同时除以x^50
=lim(x->∞) (2-1/x)^30.(3- 1/x)^20/ (2 +1/x)^50
=2^30 . 3^20 / 2^50
= (3/2)^20
lim(x->0) [√(x^2+1) -1] /[√(x^2+9) -3]
=lim(x->0) [(x^2+1) -1] /{ [√(x^2+9) -3]. [√(x^2+1) +1] }
=(1/2)lim(x->0) x^2 / [√(x^2+9) -3]
=(1/2)lim(x->0) x^2 .[√(x^2+9) +3]/ [(x^2+9) -9]
=(1/2)lim(x->0) [√(x^2+9) +3]
=6/2
=3
(2)
x^(1/3) -1 = [ x^(1/6)-1 ] [ x^(1/6) +1 ]
[ x^(1/2) -1 ] = [ x^(1/6) -1 ] [ x^(1/3) + x^(1/6) +1 ]
lim(x->1) [ x^(1/3) -1 ]/ [ x^(1/2) -1 ]
=lim(x->1) [ x^(1/6)-1 ] [ x^(1/6) +1 ]/ { [ x^(1/6) -1 ] [ x^(1/3) + x^(1/6) +1 ] }
=lim(x->1) [ x^(1/6) +1 ]/ [ x^(1/3) + x^(1/6) +1 ]
=2/3
(3)
lim(x->∞) { √[(x+2)(x-1)] -x }
=lim(x->∞) [(x+2)(x-1) -x^2 ]/ { √[(x+2)(x-1)] +x }
=lim(x->∞) (x-2)/ { √[(x+2)(x-1)] +x }
=lim(x->∞) (1- 2/x)/ { √[(1+2/x)(1-1/x)] +1 }
= 1/( 1+1)
=1/2
(4)
lim(x->∞) (2x-1)^30.(3x-1)^20/ (2x+1)^50
分子分母同时除以x^50
=lim(x->∞) (2-1/x)^30.(3- 1/x)^20/ (2 +1/x)^50
=2^30 . 3^20 / 2^50
= (3/2)^20
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