Calculus problems

A rational function has a vertical asymptote at $x = a$ and a hole at $x = b$. Which of the following statements must be true regarding the function's form?

Consider two vectors, $\vec{u}$ and $\vec{v}$, in three-dimensional space. Which of the following conditions implies that $\vec{u}$ and $\vec{v}$ are orthogonal (perpendicular)?

Given the parametric equations $x = 3\cos(t)$ and $y = 4\sin(t)$, which of the following rectangular equations represents the same curve?

A function is defined as $f(x) = \frac{ax + b}{cx + d}$, where $a$, $b$, $c$, and $d$ are constants. What condition must be met to ensure that the function has an inverse?

Consider the logarithmic equation $\log_a(x) + \log_a(x-2) = 1$, where $a > 0$ and $a \neq 1$. Which of the following statements is true about the solution(s) for $x$?

Given the polar equation $r = 2\cos(3\theta)$, which of the following best describes the graph?

Which of the following transformations will result in the graph of $y = 2^{x+1} - 3$?

A recursive sequence is defined as $a_1 = 2$ and $a_{n+1} = 3a_n - 1$ for $n \geq 1$. What is the value of $a_4$?

Which statement regarding the limit $\lim_{x \to 0} \frac{\sin(ax)}{bx}$, where $a$ and $b$ are non-zero constants, is most accurate?

A trigonometric function is given by $f(x) = A\cos(Bx + C) + D$. How does the constant $C$ affect the graph of the function?