1.for which pair(s) of functions will lim x->oo f(x)/g(x)=0?(请详解)f(x) g(x)a) e^x x^2b)e^ x In xc) In x e^xd)x In xe)3^ x 2^ x2let f(x)=xsinx.find an equation for the line tangent to df(x)/dx at x= 派(-)/5( 儿)

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1.forwhichpair(s)offunctionswilllimx->oof(x)/g(x)=0?(请详解)f(x)g(x)a)e^xx^2b)e^xInxc)Inxe^xd)xInxe)3^x

1.for which pair(s) of functions will lim x->oo f(x)/g(x)=0?(请详解)f(x) g(x)a) e^x x^2b)e^ x In xc) In x e^xd)x In xe)3^ x 2^ x2let f(x)=xsinx.find an equation for the line tangent to df(x)/dx at x= 派(-)/5( 儿)
1.for which pair(s) of functions will lim x->oo f(x)/g(x)=0?(请详解)
f(x) g(x)
a) e^x x^2
b)e^ x In x
c) In x e^x
d)x In x
e)3^ x 2^ x
2let f(x)=xsinx.find an equation for the line tangent to df(x)/dx at x= 派(-)/5
( 儿)

1.for which pair(s) of functions will lim x->oo f(x)/g(x)=0?(请详解)f(x) g(x)a) e^x x^2b)e^ x In xc) In x e^xd)x In xe)3^ x 2^ x2let f(x)=xsinx.find an equation for the line tangent to df(x)/dx at x= 派(-)/5( 儿)
工科留学生为你解答.
1.可以观察到当x->oo,f(x)和g(x)都趋向于正无穷,写成比的形式就可以用洛必达法则(L'Hopital's Role)求极限,我们一个个来看:
a) 用两次洛必达法则,分子为e^x,分母为2,当x->oo时,它们的比值趋近于正无穷而不是0,排除;
b)用一次洛必达法则,分子为e^x,分母为1/x,当x->oo时,是一个趋近于正无穷的数比上一个趋近于0的数,比值趋近于正无穷而不是0,排除;
c) 与b)正好是分子与分母进行了交换,是一个趋近于0的数比上一个趋近于正无穷的数,比值趋近于0,正确;
d)用一次洛必达法则,分子为1,分母为1/x,当x->oo时,比值趋近1/0,即正无穷而不是0,排除;
e)f(x)/g(x)=(3/2)^x,无需用洛必达法则,只要知道指数函数当底数大于1时(这里是3/2),函数值在自变量x趋向于正无穷时也趋向于正无穷,而不是0,排除.
综上,只有c)满足条件.
*以后只要知道函数增长快慢的顺序即可:对数函数(logarithmic function)

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