The year is 3500 and your spaceship has crashed on the planet Fluxion. Fortunately, your computer has survived the impact,and you open it up to determine the next course of action. You learn that your computer has some maps and information on the civilization, and that the city (and perhaps the planet) was abandoned several years ago. Based on the maps, it appears that your only hope of reaching a communication device is to head to what appears to be the center of a city that you crashed near. Your computer warns you that the city is protected by several gates, which have passcodes to activate.
The maps and images show a strange alien language that you have never encountered.
You ask your computer if it can translate this
language and are told that the language, based on its structure, can only be translated into mathematics. The computer will translate the gates’ riddles and any other information into mathematics. You will need to answer the riddle as posed by your computer (in mathematical language) for it to be able to enter the passcode for the gates. Fortunately, you have a notebook, pencil, and a mathematics text on your computer as supports. You take your supplies, take one last look at your broken ship, and head to towards the first gate.
The First Gate
Upon arriving at the first gate, you ask your computer to translate the riddle. “These appear to be limits”, your computer replies. “If you provide me with the answer to each of below questions, I will compile, translate, and communicate the passcode to the gate. There is note here that the parameter
represents a real number such that
.” Your computer also reminds you to type “infinity” for
, “-infinity” for
, and “NA” if the limit does not exist. You grab your notebook and pencil and begin

The Second Gate
As you move through the first gate, you can see another gate not far in front of you. You approach the second gate and your computer reads: “There are 2 values for which the below function does not exist. The passcode is the limit of
as
approaches the smaller of these two values.”
Your computer also reminds you to type “infinity” for
, “-infinity” for
, and “NA” if the limit does not exist.
What do you enter for your computer to translate?

The Third Gate
You arrive at the third gate and look for the inscription. Your computer translates the following:
“This passcode consists of the two below limits which include an arbitary real number,
. Remember to type “infinity” for
,”-infinity” for
, and “NA” if the limit does not exist. Enter the results and I will apply the passcode.”

The Fourth Gate
You arrive at the fourth gate, feeling confident in your skills. “It is a good thing I had some practice before landing on this planet”, you think. You look carefully at the gate and find the following information, translated by your computer:
Each of the limits below represent a piece of the passcode that you must enter to proceed. The parameters
and
represent positive real numbers and should be treated as such in your answer.
“Remember,” says your computer, “to type “infinity” for
, “-infinity” for
, and “NA” if the limit does not exist.”

The Map
As you walk through Gate 4, you realize that Gate 5 is nowhere to be found. In front of you lies an expansive desert as far as you can see. You look at your map and see that there is an alert for this area. The warning states that this desert is almost entirely quicksand, with only one path safely through the desert. That path is defined by a piecewise function but requires that some parameters be determined before your computer can generate the plot. Thinking back on what you learned in calculus, you realize that your path will need to be continuous. You just need to tell your computer what the parameters below should be in order to create this continuous path. What parameters do you give your computer?

The Fifth Gate
After traversing your continuous path, you finally arrive at Gate 5. Your computer reads the gate’s directions to you. “The passcode for this gate is the number found by first entering
and then entering
using the definitions below. ”What values do you provide to your computer?

The Sixth Gate
After walking through the fifth gate, your computer informs you that more than half of the gates have been crossed. “Only 4left”, you conclude. “Computer, what is this gate’s riddle?”

The Seventh Gate
As you approach the seventh gate, you notice that it is only a short distance to the area where your computer indicated the communication device was stored. You look past the gate hopefully, as your computer translates:

The Eighth Gate
Arriving at the eighth and penultimate gate, you immediately get to work. “Computer, translate this gate for me.” Your computer replies with the follow:
The function
models the position of an oscillating particle that has recently been discovered. Assume that
is a constant. What is the function that represents the velocity of this particle?
Note: If you want to write a power of a trigonometric function, remember that writing sin(x) to the fourth power as
is just a shorthand method. To be mathematically correct, you should write that as
– which is what your computer expects and can translate.

The Ninth Gate
You arrive at the final gate, which leads into a building-like structure. You hope that the communication device is past this last wall. You notice that there are tiles on this gate that can be moved and sorted. Your computer translates the gate with the following:
Order the tiles below from least to greatest. The smallest value must go to the left end and the largest value must go on the right end. Assume that
is a constant and that
. Use the function definitions provided in organizing the tiles.

Aiming the Communication Device
You reach the communication device and are alerted by your computer that the communication device is surrounded by aforce field that can be described by the equation,
, with the origin being where youlocated this device. The inhabitants of this planet did not want unwelcomed guests, so they made sure to guard the use ofthis device. Your computer helpfully generates a plot of this equation and tells you that it is a cardioid (whatever that means).The communication device can be taken to the very edge of the force field, but not beyond it. In fact, the only way to use thedevice is to align it tangent to the force field. Your computer also recovers some files that give points along the force fieldwhere the communication device can be used to communicate with different planets. Communication with your home planetcan be achieved from the point on the force field,
. If you provide your computer with the equation of the line that satisfies these conditions, you will be able to send a message to your home world. What equation do you provide your computer?