Question

determine the following limit at infinity. (limlimits_{x \to infty} \frac{5 + 7x + 9x^2}{x^2}) select the correct choice below and, if necessary, fill in the answer box to complete your choice. (\bigcirc) a. (limlimits_{x \to infty} \frac{5 + 7x + 9x^2}{x^2} = square) (\bigcirc) b. the limit does not exist and is neither (-infty) nor (infty).

Explanation:

Step1: Split the fraction

Split the numerator terms over the denominator: $\lim_{x\to\infty} \frac{5}{x^2} + \frac{7x}{x^2} + \frac{9x^2}{x^2}$

Step2: Simplify each term

Simplify each fraction: $\lim_{x\to\infty} \frac{5}{x^2} + \frac{7}{x} + 9$

Step3: Evaluate limits at infinity

As $x\to\infty$, $\frac{5}{x^2}\to0$ and $\frac{7}{x}\to0$. So the limit becomes $0 + 0 + 9$.

Answer:

A. $\lim\limits_{x\to\infty} \frac{5 + 7x + 9x^2}{x^2} = 9$