Evaluate the limit \( \lim _{x \rightarrow \infty} \frac{11 x^{2}-2 x+6}{4 x+8} \)

To evaluate the limit \[ \lim_{x \rightarrow \infty} \frac{11x^{2} - 2x + 6}{4x + 8}, \] let's analyze the behavior of the numerator and the denominator as \( x \) approaches infinity. 1. **Degree Comparison:** - **Numerator:** The highest power of \( x \) is \( x^2 \), so the degree is 2. - **Denominator:** The highest power of \( x \) is \( x \), so the degree is 1. 2. **Dominant Terms:** - As \( x \) becomes very large, the lower-degree terms (\( -2x + 6 \) in the numerator and \( 8 \) in the denominator) become negligible compared to the highest-degree terms. - Therefore, the expression behaves like: \[ \frac{11x^{2}}{4x} = \frac{11}{4}x \] 3. **Limit Evaluation:** - The simplified expression \( \frac{11}{4}x \) clearly tends to \( \infty \) as \( x \) approaches \( \infty \). Therefore, the original limit evaluates to positive infinity. **Answer:** The limit is \( +\infty \).