Question 12 (5 points) Evaluate \( \lim _{x \rightarrow \infty} \frac{4 x^{2}+3 x-7}{\sqrt{2 x^{2}-11 x+12}} \). If the corresponding limit is infinity, then insert INF in the blank. If the corresponding limit is negative infinity, then insert -INF in the blank. If the corresponding limit does not exist, then insert DNE in the blank. Otherwise, write the number as an integer or a reduced fraction. INF A/

To evaluate the limit \( \lim _{x \rightarrow \infty} \frac{4 x^{2}+3 x-7}{\sqrt{2 x^{2}-11 x+12}} \), we start by simplifying the expression. As \( x \) approaches infinity, the leading term in both the numerator and the denominator will dominate the behavior of the function. 1. For the numerator, \( 4x^2 + 3x - 7 \), the leading term is \( 4x^2 \). 2. For the denominator, \( \sqrt{2x^2 - 11x + 12} \), when simplified under large \( x \) limits, behaves like \( \sqrt{2x^2} = \sqrt{2}x \). Now we rewrite the limit focusing on the dominant terms: \[ \lim_{x \to \infty} \frac{4x^2}{\sqrt{2}x} = \lim_{x \to \infty} \frac{4x}{\sqrt{2}} = \frac{4}{\sqrt{2}}x. \] As \( x \to \infty \), this limit clearly tends to infinity. Therefore, we conclude: The answer is INF.