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13 problems match “limits”. 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. Continuity constant for a piecewise function ●○○○○ 10 ptsLet $f(x)=\begin{cases} kx^{2} & \text{if } x\le 2 \\ x+k & \text{if } x>2\end{cases}$. Find the value of the constant $k$ that makes $f$ continuous. Round to two decimal places.
5. 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.
6. Limit of nested square roots ●●○○○ 20 ptsFind $\lim_{x\to\infty}\left(\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}\right)$.
7. Limit of quotient of root differences ●●○○○ 20 ptsFind $\lim_{x \to 0} \frac{(1+x)^{1/2} -1}{(1+x)^{1/3} -1}$ without using L'Hôpital's rule. Round to three decimal places.
8. Continuity with parameters a and b ●●○○○ 20 ptsLet $f,g:\mathbb{R}\to\mathbb{R}$ be the functions $$f(x)=\begin{cases}-x+3,&x\leq 1,\\-ax^2,&\text{otherwise},\end{cases}\qquad g(x)=\begin{cases}\dfrac{x^3+x^2-x-1}{x-1},&x\neq 1,\\-b,&\text{otherwise}.\end{cases}$$ Determine the parameters $a,b\in\mathbb{R}$ such that $f$ and $g$ are both continuous. Find $a$ and $b$.
9. 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.
10. Minimum roots of g times g' ●●●○○ 50 ptsLet $f$ be a function defined for all $x \in \mathbb{R}$ that is discontinuous at exactly $3$ values of $x$. Let $g$ be a differentiable function such that $f(x)g(x)$ is continuous for all $x \in \mathbb{R}$. Find the minimum number of distinct real roots of the equation $g(x)g'(x)=0$.

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