Introduction

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 hassome 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 nearby. 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 thislanguage 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 parametera?represents a real number such that0

limlim\f