Question

请问求极限这么化简对吗

求解如图，有没有替换不能用的情况

Answer 1

x->0
e^x = 1+x+o(x)
e^(x/2)=1+x/2+o(x)
[e^x +e^(x/2)]/2 = 1+ (3/4)x +o(x)
lim(x->0) { [e^x +e^(x/2)]/2 }^(1/x)
=lim(x->0) [ 1+(3/4)x ] ^(1/x)
=e^(3/4)

Answer 2

=e^limln((e^x+e^(x/2))/2)/x=e^lim((e^x+e^(x/2))/2－1)/x=e^lim(e^x－1)/2x+(e^(x/2)－1)/2x=e^(1/2+1/4)=e^(3/4)

Answer 3

x->0e^x = 1+x+o(x)e^(x/2)=1+x/2+o(x)[e^x +e^(x/2)]/2 = 1+ (3/4)x +o(x)lim(x->0) { [e^x +e^(x/2)]/2 }^(1/x)=lim(x->0) [ 1+(3/4)x ] ^(1/x)=e^(3/4)

Answer 4

=e^limln((e^x+e^(x/2))/2)/x
=e^lim((e^x+e^(x/2))/2－1)/x
=e^lim(e^x－1)/2x+(e^(x/2)－1)/2x
=e^(1/2+1/4)
=e^(3/4)

Answer 5

求高数极限化简解释 lim(x0) sin³mx / x²=lim(x0) (sinmx/mx)(sinmx/mx) m²sinmx利用重要极限：lim(x0) (sinx/x) =1所以，原式 =lim(x0) 1·1·sinmx ·m²=lim(x0) (sinmx/mx) m³x不知道你有没有学过等价无穷小：当 mx趋于0时，sinmx等价于mx原式 = lim(x0) sin³mx / x²= lim(x0) (mx)³ / x²= m³x = 0 这样做简单很多