Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
\[
\lim _{x \rightarrow \infty} \frac{6 x^{2}+2 x}{9 x^{2}-x+1}
\]
Select the correct choice below and, if necessary, fill in the answer box within your choice.
A. $\lim _{x \rightarrow \infty} \frac{6 x^{2}+2 x}{9 x^{2}-x+1}=$
(Simplify your answer. Type an integer or a fraction)
B. The limit does not exist and is neither $\infty$ nor $-\infty$.
Answer
So, the final answer is \(\boxed{\frac{2}{3}}\).
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 \(\boxed{\frac{2}{3}}\).
