3. Suppose that r(t) < s(t) on [1, ), and r(t) is continuous and strictly decreasing on [1, 0). If Ss(t)dt is converge, then lim r(n) = 0. m

Q: 7. A function f : (a,6) → R satisfies a Lipschitz condition at x € (a,6) iff there is M > 0 and e >…

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Q: Express a = the limit lim Σ(4(x*)² – 3(x* )³) Aï; over [6, 7] as an integral. n→∞ i=1 Provide a, b…

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Q: 7. Suppose that f: R → R is continuous at c and that f(c) 0 such that f(x) < 0 in the…

A: Recall the definition of continuity around c .And then choose an ∈&gt;0 to get a δ&gt;0…

Q: True or false: If f: R\ {0} → R is uniformly continuous, then lim f(x) 2-0 exists. True False

A: f: ℝ\0→ℝ is a uniform continuous function , We have to verify whether limx→0fx exist or not

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Q: 6. Let f: Q R be defined by f(x) Show that f is continuous. 0, :{ 1, = if x € (-∞, √2) nQ if x €…

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Q: Show that a continuous function f : [0, 1] → Z is a constant function.

A: Consider a function .We need to show that a continuous function  is a constant function.We know…

Q: 6. Let f(x) = x2/3. (a) Then fis uniformly continuous on [-1,1]. Why? (b) Show that there does not…

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Q: If ƒ is continuous for a < x < b and differentiable for a < x < b , which of the following could be…

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Q: 25. Suppose f: (a, b) → R is differentiable and f'(x)| ≤ M for all x E (a, b). Prove that f is…

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Q: Suppose f(z) is defined on A = {z Re(2) >0} such that f(z) is differentiable at each point in A and…

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Q: Prove that if m(x) is differentiable on (−∞,∞) and its derivative m'(x) is bounded then m is…

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