Question 1044770
C){{{e^5}}}.

{{{lim(x->infinity, ((x+6)/(x+1))^(x+4))}}}

= {{{lim(x->infinity, ((1 + 5/(x+1))^((x+1)/5)))^5*(1+5/(x+1))^3}}}

= {{{lim((x+1)/5->infinity, ((1 + 5/(x+1))^((x+1)/5)))^5*(1+5/(x+1))^3}}}

Let {{{n = (x+1)/5}}}

===> {{{lim(n->infinity, ((1 + 1/n)^n))^5*(1+1/n)^3}}}

= {{{lim(n->infinity, ((1 + 1/n)^n))^5*lim(n->infinity,(1+1/n)^3)}}}

= {{{(lim(n->infinity, (1 + 1/n)^n))^5*lim(n->infinity,(1+1/n)^3)}}}

= {{{e^5*1^3}}}

= {{{e^5}}}, using the fact that {{{lim(n->infinity, (1+1/n)^n )= e}}}.