Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $f(x)=\frac{x^{2}-2}{x^{2}-4}$

Let $n \in \mathbb{Z}, n>1,$ and let $a \in \mathbb{Z}$ with $1 \leq a \leq n$. Prove if a and n are not relatively prime, there exists an integer b with $1 \leq b<n$ such that $a b \equiv 0(\text{mod } n)$ and deduce that there cannot be an integer c such that $a c=1(\text{mod } n)$.

Game tree

The capacity of any cut is greater than or equal to the value of any flow.

Represent the postfix expression as a binary tree and write the prefix form, the usual infix form, and the fully parenthesized infix form of the expression. AB+C−

a. Using semilog paper, plot $X_{L}$ versus frequency for a 10 mH coil and a frequency range of 100 Hz to MHz. Choose the best vertical scaling for the range of $X_{L}$. b. Repeat part (a) using log-log graph paper. Compare to the results of part (a). Which plot is more informative? c. Using semilog paper, plot $X_{C}$ versus frequency for a 1 $\mu$ F capacitor and a frequency range of 10 Hz to 100 kHz. Again choose the best vertical scaling for the range of $X_{C}$. d. Repeat part (a) using log-log graph paper. Compare to the results of part (c). Which plot is more informative?

A certain bacterium is known to grow according to the Malthusian model, doubling itself every 8 hours. If a biologist starts with a culture containing 20,000 bacteria, then harvests the culture at a constant rate of 2000 bacteria per hour, how long until the culture is depleted? What would happen in the same time span if the initial culture contained 25.000 bacteria?

Instant Insanity

The lattice energy of KF is 794 kJ/mol, and the interionic distance is 269 pm. The Na–F distance in NaF, which has the same structure as KF, is 231 pm. Which of the following values is the closest approximation of the lattice energy of NaF: 682 kJ/mol, 794 kJ/mol, 924 kJ/mol, 1588 kJ/mol, or 3175 kJ/mol? Explain your answer.

Encode each word using the preceding Huffman code. PENNED

Sketch the Bode plot of the following function:

$$\mathbf{A}_{v}=\frac{j f / 1000}{(1+j f / 1000)(1+j f / 10,000)}$$

Draw a graph having the given properties or explain why no such graph exists. Tree; all vertices of degree 2

The following problem introduce the idea-developed more fully in the next section-of a multiplicative inverse of a square matrix. Use the technique of earlier problem to show that if

$$\mathbf{A}=\left[\begin{array}{rr} 1 & -2 \\ -2 & 4 \end{array}\right]$$

, then there does not exist a matrix B such that AB = I.

Use the substitution $u(r, t)=v(r, t)+\psi(r)$ to solve the given boundary-value problem. $\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\beta=\frac{\partial u}{\partial t}, \quad 0<r<1, \quad t>0, \quad \beta$ a constant

$$\begin{array}{ll} u(1, t)=0, & t>0 \\ u(r, 0)=0, & 0<r<1 \end{array}$$

.