a) Let f(x) = x cos(x), g(x) = x + cos(x). Find f(x) lim I-00...

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a) Let f(x) = x – cos(x), g(x) = x + cos(x). Find f(x) lim I-00 g(x) without using l'Hopital's rule. What happens if you try to use l'Hopital's rule? Explain. b) Let f(x) = exp(-22)(cos(x) + 2 sin(x)), g(2) = exp(-2)(cos(x) + sin(x)). Show that f(x) lim 2-00 g(x) does not exist. What happens if you try to use l'Hopital's rule? Explain. Note: We didn't introduce the exponential function exp: R R yet; the main properties, which you may use without proof, are exp (ӕ) — еxp(2), еxp(а + b) — еxp(а) ехp(b), lim exp(x) = ∞ Note: For two functions f, g: A → R, we take the domain of f/g to be the set of all x € A such that g(x) 0. a) Let f(x) = x – cos(x), g(x) = x + cos(x). Find f(x) lim I-00 g(x) without using l'Hopital's rule. What happens if you try to use l'Hopital's rule? Explain. b) Let f(x) = exp(-22)(cos(x) + 2 sin(x)), g(2) = exp(-2)(cos(x) + sin(x)). Show that f(x) lim 2-00 g(x) does not exist. What happens if you try to use l'Hopital's rule? Explain. Note: We didn't introduce the exponential function exp: R R yet; the main properties, which you may use without proof, are exp (ӕ) — еxp(2), еxp(а + b) — еxp(а) ехp(b), lim exp(x) = ∞ Note: For two functions f, g: A → R, we take the domain of f/g to be the set of all x € A such that g(x) 0.