lim(x+2^x)^(1/x) x趋向于0这道题怎么做

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lim(x+2^x)^(1/x) x趋向于0这道题怎么做
lim(x+2^x)^(1/x) x趋向于0
这道题怎么做

lim(x+2^x)^(1/x) x趋向于0这道题怎么做
ln[(x+2^x)^(1/x)]=ln(x+2^x)/x,利用L'Hospital法则,得limln(x+2^x)/x=ln2+1.所以最后结果为2e

lim (x+2^x)^(1/x)
= lim e^((1/x)ln(x+2^x))
= e^(lim (ln(x+2^x) / x))
= e^(lim (1+2^x ln2)/(x+2^x) / 1)
= e^(lim (1+2^x ln2)/(x+2^x))
= e^(lim (1+ln2)/(0+1))
= e^(1+ln2)
= 2e.

原式=lim{x-->0}(1+x+2^x-1)^{[1/(x+2^x-1)]*[(x+2^x-1)/x]}
=lim{x-->0}{(1+x+2^x-1)^[1/(x+2^x-1)]}^[(x+2^x-1)/x]}
lim{x-->0}(1+x+2^x-1)^[1/(x+2^x-1)]=e
lim{x-->0}[(x+2^x-1)/x]=lim{x-->0}[1+(2^x-1)/x]=1+ln2
所以原式=e^(1+ln2)=2e

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