Step 1 :The given limit is \(\lim _{x \rightarrow \infty} \frac{6 x^{2}+2 x}{9 x^{2}-x+1}\).
Step 2 :This is a ratio of two polynomials, and both the numerator and the denominator are of the same degree (2).
Step 3 :In such cases, the limit as x approaches infinity is the ratio of the leading coefficients.
Step 4 :The leading coefficients are 6 and 9, respectively.
Step 5 :Therefore, the limit should be \(\frac{6}{9}\) or \(\frac{2}{3}\).
Step 6 :So, the final answer is \(\boxed{\frac{2}{3}}\).