Decide which of the following statements are true and which are false. Prove the true ones...

Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones.

a) If f (x) → ∞as x → ∞and g(x) > 0, then g(x)/ f (x) → 0 as x →∞.

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b) If f (x) → 0 as x → a+ and g(x) ≥ 1 for all x ∈ R, then g(x)/ f (x) → ∞as x → a+.

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c) If f (x)→∞as x →∞, then sin(x2 + x + 1)/ f (x) → 0 as x →∞.

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d) If P and Q are polynomials such that the degree of P is less than or equal to the degree of Q (see Exercise), then there is an L ∈ R such that

Exercise

This exercise is used many places. Recall that a polynomial of degree n is a function of the form

where a j ∈ R for j = 0, 1, . . . , n and an ≠ 0.

a) Prove that if 00 = 1, then limx→a xn = an for n = 0, 1, · · · and a ∈ R.

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b) Prove that if P is a polynomial, then

for every a ∈ R.