7. Let a be a nonzero real number. Evaluate lim,→ ∞ f(x) and lim-→f (x), wheref(x) = ax³ – 3xª + 6x³- 7x? + 10.State a result that guarantees the existence of a real root of the polynomial f(x) given the limits above.What would go wrong in this argument if we replace f(x) with a polynomial of an even degree? 1. Let a be a positive real number and n be a natural number. Define S = {x € R : x >0 and x" < a}.(i) Prove that S is nonempty and bounded above. Let a = sup S.(ii) Assuming that a" < a, show that for some natural number m1, the inequality (a + 1/m1)" < aholds. Why does this show that a" > a?(iii) Assuming that a" > a, show that for some natural number m2, the inequality (a – 1/m2)" > aholds. Why does this show that a" < a?(iv) Conclude from (ii) and (iii) above the existence of a positive nth root of a. Is such positive nthroot of a unique? If yes then prove your claim, otherwise give a counterexample.

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Let a be a nonzero real number. Evaluate limr→ ∞ f(x) and lim,→ -o f(x), where -00 f(x) = ax³ – 3x4 + 6x³ 7x2 + 10. State a result that guarantees the existence of a real root of the polynomial f(x) given the limits above. What would go wrong in this argument if we replace f(x) with a polynomial of an even degree?