Math 152-02 Lab #1, January 28, 2009 Michael Pelsmajer and Allen Flavell
Try to finish this by the end of class, or Friday. The due date isTuesday at 5pm, so that in case you run into trouble, you have
Monday and Tuesday to get help. If you haven't already, save this file as "[your-name]-Jan28lab".
Save often! When you are done, revise your work so that it is easy for Allen to find and understand your solutions. (Also, for yourself, since you may need to come back to your earlier work.)restart;1. Graphing with technology is great! Except when it goes horribly wrong! The purpose of this exercise is to demonstrate that plots can be very misleading, so you can never simply trust what you see on a computer or calculator. You need to combine mathematical thinking with the use of technology - with both together, you can do really great stuff. 1.a. Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg== = 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. Plot LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGNEY3Rj5GPkY+RitGPg== over the interval [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 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJSZwaTtGJy8lJ2l0YWxpY0dGPkY1RjVGK0Y1 ]. (Remember to use Ctrl-M before typing next to the red ">".)1.b. From the above plot, the graph appears to have a horizontal tangent line at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRJyYjOTYwO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZHLyUpYmV2ZWxsZWRHRjQvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGNQ== which would mean that the derivative of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy9GM1Enbm9ybWFsRic= at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRJyYjOTYwO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZHLyUpYmV2ZWxsZWRHRjQtRi82I1EhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGNQ== equals...what?1.c. Use Maple to calculate the actual derivative of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy9GM1Enbm9ybWFsRic= at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRJyYjOTYwO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZHLyUpYmV2ZWxsZWRHRjQtRi82I1EhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGNQ== . (The answers to 1.b. and 1.c. should be quite different.)We'll resolve this mystery in two ways, algebraically, and then graphically.1.d. Use Maple to get a formula for the derivative of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy9GM1Enbm9ybWFsRic= (at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy9GM1Enbm9ybWFsRic= ). Use the formula to explain why 1.c. is right (and 1.b. is wrong). 1.e. Make the plot again, but change the domain to be a small interval containing LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRJyYjOTYwO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZHLyUpYmV2ZWxsZWRHRjQtRi82I1EhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGNQ== . (The following may help.)evalf(Pi/2); (It should be clear from your plot that the tangent line at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRJyYjOTYwO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZHLyUpYmV2ZWxsZWRHRjQtRi82I1EhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGNQ== is not necessarily a horizontal line, and we should get a sense of what the plot really looks like near 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.)2. The curious expression, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiK0YnRjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTy1JJm1mcmFjR0YkNigtRiM2JEYzRjctRiM2JC1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GOFEnaXRhbGljRidGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYW8vJSliZXZlbGxlZEdGQEY3RjdGWS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGNw==. As LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoJmluZmluO0YnRjRGN0Y+RitGPg==, the inner part gets very close to 1, and the exponent grows huge (i.e., "goes to infinity"). For short, people often write this as 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; this is an "indeterminate form", which is to say that the value of the limit isn't obvious. Let's explore that - what's a reasonable way to try to figure out the answer, without working too hard? - You might think that it behaves like 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJ0Y7Rj4vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjI= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yv, since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRidGO0Y+LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZFLyUpYmV2ZWxsZWRHUSZmYWxzZUYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjI= becomes very close to 0 as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoJmluZmluO0YnRjRGN0Y+RitGPg==. - You might think that it behaves like 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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEmMS4wMDFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRidGO0Y+LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Yy, since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRidGO0Y+LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZFLyUpYmV2ZWxsZWRHUSZmYWxzZUYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjI= becomes small but never reaches 0, as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoJmluZmluO0YnRjRGN0Y+RitGPg==.Might this reasoning be good enough?It's good enough if both functions have the same limit, because 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 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYkLUYjNigtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiK0YnRjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTy1JJm1mcmFjR0YkNihGMy1GIzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnL0Y4USdpdGFsaWNGJ0ZlbkZobi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGX28vJSliZXZlbGxlZEdGQC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0Y3RjdGVS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGZG9GZ29GNw== < 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 (when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy9GM1Enbm9ybWFsRic= is large enough),so the Squeeze Theorem would imply that all the limits are the same. Unfortunately, it doesn't work.2.a. Plot the function 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 on an interval that includes some rather large x-values, so that we can see that the limit at infinity is definitely not 1, and that it appears like the limit might actually be infinity.Okay. In this case - actually, more generally, for any limit with theindeterminate form 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- we now know that the value of the limit is NOT AT ALL OBVIOUS.Fine. Now that we know that it's not easy, we should be willing to work a little harder to find the answer... and in fact, during class today you saw a trick which allows you to find this limit.(You remember the answer, right?)2.b. Put larger and larger numbers into 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 check whether it's getting close to the known value of 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.c. Use the same experimental technique to try to figure out the value of 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What does it seem to be?2.d. (OPTIONAL/BONUS) Prove that your guess in 2.c. is correct. Hint: Start with 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, and the correct value that it equals,and substitute 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 . (The rest you have to figure out.)