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6 problems match “limits at infinity”. Clear

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1. Horizontal asymptote of a rational function ●○○○○ 10 ptsAt what value of $y$ is there a horizontal asymptote for the graph of the equation $y=\frac{4x^3+2x-4}{3x^3-2x^2+5x-1}$? Round to two decimal places.
2. Limit of binomial coefficient over quadratic ●○○○○ 10 ptsCalculate $\lim_{n \to \infty} \frac{\binom{n}{2}}{n^2+1}$. Give your answer as a decimal.
3. Limit with sine of factorial ●○○○○ 10 ptsEvaluate $\lim_{n\to\infty}\frac{n^{1/3}\sin(n!)}{1+n}$.
4. Limit of a square-root difference ●●○○○ 20 ptsFind $\lim_{n\to\infty} \sqrt{n}\left(\sqrt{n+1}-\sqrt{n}\right)$. Give your answer as a decimal.
5. Limit of nested square roots ●●○○○ 20 ptsFind $\lim_{x\to\infty}\left(\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}\right)$.
6. Limit of xth root of factorial over x ●●●○○ 50 ptsFind the value of $\lim_{n\to\infty} \frac{\sqrt[n]{n!}}{n}$, where $n$ is a positive integer. Round to two decimal places.