lim x(√(9x^2+1) - 3x ) x趋于正无穷

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lim x(√(9x^2+1) - 3x ) x趋于正无穷
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lim x(√(9x^2+1) - 3x ) x趋于正无穷
lim x(√(9x^2+1) - 3x ) x趋于正无穷

lim x(√(9x^2+1) - 3x ) x趋于正无穷
(√(9x^2+1) - 3x )
=(√(9x^2+1) - 3x )(√(9x^2+1) +3x )/(√(9x^2+1) +3x )
=(9x^2+1-9x^2)/((√(9x^2+1) + 3x )
=1/(√(9x^2+1) + 3x )
原式=lim x/(√(9x^2+1) + 3x )
分子分同除以x
=lim1/((√(9+1/x^2) +3)
=1/(根号(9+0)+3)
=1/6

x 1
lim x(√(9x^2+1) - 3x )=lim -------------------------= lim -----------------------------=1/6
x→+∞ x→+∞ √(9x^2+1) +3x x→+∞ √(9+1/x^2) +3

lim(x→∞) x(√(9x^2+1) - 3x ) (分子有理化)
=lim(x→∞) x(√(9x^2+1) - 3x )(√(9x^2+1) + 3x )/(√(9x^2+1) +3x )
=lim(x→∞) x/(√(9x^2+1) +3x ) (看最高次项系数比)
=1/6