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1. Degree of numerator for horizontal asymptote ●○○○○ 10 ptsThe graph of the rational function $\frac{q(x)}{2x^5+x^4-7x^2+1}$ has a horizontal asymptote. What is the largest possible degree of $q(x)$?
2. Sum of vertical asymptotes ●○○○○ 10 ptsThe graph of $y=\frac{5x^2-9}{3x^2+5x+2}$ has vertical asymptotes at $x = a$ and $x = b$. Find $a + b$. Round to two decimal places.
3. 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.
4. 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.
5. Crossing the horizontal asymptote ●●○○○ 20 ptsFor which value of $x$ does the function $f(x) = \frac{2x^2 - 5x - 7}{x^2 - 4x + 1}$ cross its horizontal asymptote?
6. Positive integers giving a hole ●●○○○ 20 ptsFind the number of positive integers $n$ for which the graph of $y = \frac{(x + 3)(x - 7)^5}{(x - 1)(x - 7)^n}$ has a hole at $x = 7$.
7. Horizontal tangent of a rational function ●●○○○ 20 ptsThe graph of $f(x)=\frac{ax+b}{(x-1)(x-4)}$ has a horizontal tangent line at $(2,-1)$. Determine the values of $a$ and $b$.
8. Rational function with horizontal tangent ●●○○○ 20 ptsThe graph of $g(x) = \frac{ax+b}{(x-1)(x-4)}$ passes through the point $(2,-1)$ and has a horizontal tangent there. Find the values of $a$ and $b$.
9. 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 $b$.
10. Finding a from two points on a graph ●●●○○ 50 ptsGiven a constant $a > 0$, the graph of the function $f(x)= \frac{2^{x}}{2^{x}+ax}$ passes through the points $P\left(p, \frac{6}{5}\right)$ and $Q\left(q, -\frac{1}{5}\right)$. If $2^{p+q}=36pq$, find the value of $a$.

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