\lim_{x\rightarrow infinity}\frac{x+1}{x^{2}+x+1}=\lim_{x\rightarrow infinity}\frac{\frac{1}{x}+\frac{1}{x^{2}}}{1+\frac{1}{x}+\frac{1}{x^{2}}}=0
since \frac{1}{x}\rightarrow 0\; as\; x\rightarrow infinity
\lim_{x\rightarrow infinity}\frac{x+1}{x^{2}+x+1}=\lim_{x\rightarrow infinity}\frac{\frac{1}{x}+\frac{1}{x^{2}}}{1+\frac{1}{x}+\frac{1}{x^{2}}}=0
since \frac{1}{x}\rightarrow 0\; as\; x\rightarrow infinity