Question

大一高数求教

Answer 1

追问

第二步看不懂😂

Answer 2

lim(x->∞) ln(1+2^x). ln(1+ 3/x)

=lim(x->∞) ln(1+2^x)/[1/ ln(1+ 3/x)]                         ( ∞/∞ 分子分母分别求导)

=lim(x->∞) [ (ln2).2^x/(1+2^x) ]/{  [-1/(x+3) + 1/x]/[ln(1+ 3/x)]^2  }

=(ln2)lim(x->∞) 1/{  [-1/(x+3) + 1/x]/[ln(1+ 3/x)]^2  }

=(ln2) .lim(x->∞) x(x+3) .[ln(1+ 3/x)]^2  /3        

=(1/3)(ln2) .lim(x->∞) x(x+3) .[ln(1+ 3/x)]^2 

y= 1/x

=(1/3)(ln2) .lim(y->0) (1/y)(1/y+3) .[ln(1+ 3y)]^2 

=(1/3)(ln2) .lim(y->0) (3y+1).[ln(1+ 3y)]^2  / y^2

=(1/3)(ln2) .lim(y->0) [ln(1+ 3y)]^2  / y^2

=(1/3)(ln2) .lim(y->0) 9y^2 / y^2

=(1/3)(ln2) (9)

=3ln2