Question 839203
{{{lim(n->infinity,(sqrt(n^2+n+1)-n))=lim(n->infinity,(sqrt(n^2+n+1)-n)*(sqrt(n^2+n+1)+n)/(sqrt(n^2+n+1)+n))}}}
{{{lim(n->infinity,(sqrt(n^2+n+1)-n))=lim(n->infinity,(n^2+n+1-n^2)/(sqrt(n^2+n+1)+n))}}}
{{{lim(n->infinity,(sqrt(n^2+n+1)-n))=lim(n->infinity,(n+1)/(sqrt(n^2+n+1)+n))}}}
{{{lim(n->infinity,(sqrt(n^2+n+1)-n))=lim(n->infinity,(n(1+1/n))/((n)*(sqrt(1+1/n+1/n^2)+1))))}}}
{{{lim(n->infinity,(sqrt(n^2+n+1)-n))=lim(n->infinity,(1+1/n)/(sqrt(1+1/n+1/n^2)+1)))}}}
Now taking the limit,
{{{lim(n->infinity,(sqrt(n^2+n+1)-n))=(1+0)/(sqrt(1+0+0)+1)))}}}
{{{lim(n->infinity,(sqrt(n^2+n+1)-n))=1/(1+1)}}}
{{{lim(n->infinity,(sqrt(n^2+n+1)-n))=highlight_green(1/2)}}}