(f) If (a„) is convergent, then (a,) is monotone and bounded. (g) If f is continuous and A is open, then f'(A) is open.

Q: Give an example if f : (0,1) -> R that is bounded, continuous but not uniformly continuous on (0,1).

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Q: n' (a) Ax (b) The limits a, b The function f(x) which this limit is approximating.

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Q: Let F = <5x^2, 2(x+y)^2, 9(x+y+z)^2> Find divF (divergence)

A: First of all, consider the vector F and find F1, F2, and F3.

Q: Suppose that the sequence of functions {fn} and the function f are integrable on [a, b] and that…

A: Given that,  The functions fn and f are integrable on a,b. fn converges to f on uniformly a,b. an…

Q: 1 Let {fn} bes sequence of real nonnegative functions on R', consider the following four statements:…

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Q: limx→∞ 2*+1 + 3* - 2 -1 - 3*+2

A: To determine the value of limx→∞2x+1+3x-22x-1-3x+2, divide the numerator and the denominator of the…

Q: For each natural number n and each number x in (-1,1), define f_n(x)=sqrt((x^2)+(1/n)) and define…

A: Uniformly continuity :- Let I⊂ℝ and for each n∈ℕ, let   fn : I →ℝ be  a function then the sequence…

Q: Define f(x)=x^3 for all x. Verify the ε-δ criterion for continuity at each point x_0.

A: X3 is continuous at every point in R.

Q: Prove that limit x goes to 2  x^(3)+3 = 11 (Using epsilon-delta proof)

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Q: Use any method to evaluate the limit or show that the limit does not exist. (Use symbolic notation…

A: (.)   Given ,       we find the limit . We choose a path and prove that limit does not exist.

Q: Suppose we have the sequence of functions $f_n(x)=x^n$ defined on $[0,1],$ and suppose $f_n\to f$…

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Q: Let (an) be a sequence of real numbers that diverges to +∞, and let (bn) be bounded. Prove that lim…

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Q: Let f (x) = 1/x, for all x not equal to 0. Prove: (i) f is uniformly continuous on (1, infinity)…

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Q: Let fn(x) = n²x²(1 – x) on [0, 1]. Does lim,oo fn(x) exist for every r E [0, 1]?

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Q: n=1 2m-1 m Let {} be the 'interlaced' sequence a, b, a, b defined by c₂ = a, c. = b Show that lim…

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Q: 3. Let {gn} be a sequence of integrable functions such that gn → g a.e. with g integrable. Show that…

A: Let gn be a sequence of integrable functions. Assume that  limn→∞∫gn=∫g. Then,…

Q: F B) Define-F: (0,1)-> IR by Fex) = cos I. Does I have a limit at o? Justifying D c) Define F:…

A: Given f: (0,1) ,as x So f does not have a limit at x tends to 0.

Q: 11. Prove that if f is continuous on (a, b] and lim,a* f(x) exists, then fis uniformly continuous on…

A: The theorem is done .

Q: The graph of f is shown in the figure. 6. 4 х 14 -4 -6+ 2-

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Q: Use the definition to prove f = (x-1)/(x+1) is uniformly continuous on [0,∞).

A: Given function is fx=x−1x+1. We have to show that fx=x−1x+1 is uniformly continuous on 0,∞. Let us…