n+1 ) Construct a function f(x) such that for any integer n e Z, the integral f(r)dr is improper and convergent.

Q: . Let f (x) be a continuous function on [a, b], where a, b ∈ R and a < b. Suppose that there are…

A: To prove thatlimn→∞​yn​−xn​f(yn​)−f(xn​)​=f′(c)given that  f(x)  is differentiable at  c , we follow…

Q: kx +5 for x ≤ 2 Let fx = = x-1 for x > 2 Find the value of k for which f(x) is continuous at x=2.…

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Q: Prove that the function f(x) = Root3(x) is uniformly continuous on [−1, ∞). (Hint: consider the sets…

A: Given that fx=3x we have to prove that fx is continuous in interval [-1,∞) lets consider the set…

Q: Define f(8) in a way that extends f(s) = 3 s - 512 S s²-64 2 to be continuous at s = 8.

A: The objective of the question is to determine the value of " f(8).

Q: Prove that the function ƒ : (0, ∞) → R defined by ƒ(x) = x³/² = (√x)³ is not uniformly continuous.…

A: To Prove: The function f:0, ∞→ℝ defined by fx=x3/2=x3 is not uniformly continuous. To Use the…

Q: 3. Let fn(x) = ¹+cos(nx), x € R. Find f(x) so that fn → ƒ pointwise on R, then check whether fn → f…

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Q: (c) A non-constant function f : [0, 1] → R such that f(x) > 0 for any r e [0, 1] but | f(x)dx = 0.…

A: Let us consider a function f:0,1→ℝ such that fx≥0 for all  x∈0,1.  Suppose fxn be a sequence of…

Q: Compute the left and right Riemann sums L4 and R4, respectively for f(x) = √4x² on [-2, 2] and…

A: We can find Left and Right Riemann sums .

Q: Find all the strictly monotonic functions f: R→R such that f(x + f(y)) = f(x) + y, for all z, y € R.…

A: a. Consider the given condition, fx+fy=fx+y, for all x,y∈ℝ. Here we need to find the strictly…

Q: Let f(n) =an for all positive integers n. What conditions must be met for the Integral Test? Select…

A: If the function f(x) is positive, decreasing and continuous function for some positive integer…

Q: Let Calculate f(x) = ² Σ K=O k f(K) 2 k! where K=0,1₁.. X>0

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Q: Let a function f(x) be continuous and positive for all x > 0. What does it mean to say that the…

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Q: Q1 Prove a) by the definition of continuity b) by an e-8 argument that the function f(x)=√ is…

A: The given function is f(x)=x, and the domain is [0,∞). To Prove: The above function is continuous:…

Q: Let f be a strictly positive, continuous function such that |f(x)| < x for all real x. Does the…

A: if we have 0 ≤ g(x) ≤ h(x) on the interval [a, ∞) then if  ∫a∞h(x) dx  converges  ⇒  ∫a∞g(x) dx…

Q: Suppose fn → f and the functions fn all satisfy the Lipschit condition |fn(x) – fn(y)| < Mx – y| for…

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Q: Define f(x) = x² for all x. Verify the e-8 criterion for continuity at x = 2.

A: Given real valued function  f(x) = x2, ∀x We have to show that the given function is continuous at x…