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Suppose f(x)-> and g(x)-> as x->+inf. Find examples of functions f and g with these properties and such that:

7. Suppose f (x) - 0 and g(x) - 0 as x - +co. Find examples of functions f and g with these properties and such that: a. lim f(x ) x-+co Lg(x). = +00 Prover (choose the functions): Explainer (check the limits): f ( x ) = lim f (x) = X- +00 g (x) = lim g(x) = x-+00 f (x) g(x) = lim f (x) x-+00 g(x)b. lim [f( )] x-to Lg(x)] = -00 Prover (choose the functions): Explainer (check the limits): f (x) = lim f (x) = g (x) = lim g(x) = f (x) g(x) lim (x] x-to g(x) C. lim x-too g(x) = A, where A is any arbitrary real number that you choose yourself. Prover (choose the functions): Explainer (check the limits): f (x ) = lim f (x) = g (x) = lim g(x) = f(xx) g (x ) lim x] x-too g(x) = d. Think carefully about what the symbol 0 represents in the statements f (x) - 0 and g(x) - 0. (Hint: Is it a fixed value, or does it describe a particular behavior?) Now use your work from parts a-c to explain why - is an indeterminate form. . Explain what happens when you divide any fixed number by a series of smaller and smaller numbers that become closer and closer to zero. How does the result change? Use this information to explain why the form - (where a is any fixed number) is a determinate form.f. Thinking carefully again about the meaning of the symbol zero in the statements f(x) > 0 and g(x) > O, explain how the indeterminate form 3 is different from the determinate form 3 (where a at 0)