The year is 3500 and your spaceship has crashed on the planet Fluxion. Fortunately, your computer...

   

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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 a represents a real number such that 0 < a < 5." 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. lim x→a lim x→a+ x-5 x-a x-5 x-a lim x-5 xa x-a = = = AGI Explain, in your own words and with your own work, how you arrived at this result. Be sure to explain using calculus concepts to best support the work of the game design team. 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 g (x) as a approaches the smaller of these two values." g(x) = x+3 2-9 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? 101 151 Explain, in your own words and with your own work, how you arrived at this result. Be sure to explain using calculus concepts to best support the work of the game design team. 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 a represents a real number such that 0 < a < 5." 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. lim x→a lim x→a+ x-5 x-a x-5 x-a lim x-5 xa x-a = = = AGI Explain, in your own words and with your own work, how you arrived at this result. Be sure to explain using calculus concepts to best support the work of the game design team. 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 g (x) as a approaches the smaller of these two values." g(x) = x+3 2-9 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? 101 151 Explain, in your own words and with your own work, how you arrived at this result. Be sure to explain using calculus concepts to best support the work of the game design team.