3. Assume that f is uniformly continuous on a bounded set S, prove that f(S) is bounded. (hint: thm 19.4, 11.5) 4. Let ƒ(x) = (x-1)(x−3)² , ‚ determine limä→1 ƒ (x), limx→2 ƒ (x), limx→3 ƒ (x). (hint: limit may be infinite or does not exist).

Q: x+2 Q1I) If f(x) =* and h(x) = v2 –x, 3-x find (1) (f o h)(x) (2) Domain and Range for all f (x),…

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Q: 2. Use the ɛ-d definition of the limit of a function to prove that lim = 3. I-»3 T – 2

A: Let  I  be an open interval containing  c , and let  f  be a function defined on  I , except…

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Q: 6. let (Sa) C Cla, b). Suppose there exsits a function g € Lla, b) with fa(#) S g(x) a.e. for each…

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Q: 7. Intro To Real Analysis

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A: Definition of limit: Let be a function. If for every there exists such that and we can write it…

Q: If it is continuous, there are 3 conditions needed to be satisfied: i. f(a) is defined ii. lim x→a…

A: The three criterias that you have mentioned are for a point. When we check the continuity at a…

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A: We need to show that the function does not have limit at point (0,0)

Q: Evaluate the limit using L’Hôpital’s Rule.

A: Given limit: limx→-4 2x2+13x+20x+4

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Q: Give an example of a limit of the form ∞ / ∞ as x→ 0.

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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: 20. Let f be a function of three variables that is continuous everywhere. (a) Explain why the…

A: Givenf(x,y,z) is continuous To show(a) f(x,y,z)x+y+z is continuous if and only if…

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

Q: 13. 0.) Compute limit fix, if it exists, for the following f: IR→IR ₁ × → 1×1₁ X ₁ = 1 x cases!

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Q: prove the given limit does not exist xy + yz²+xz² lim (x,y,z)+(0,0,0) x² + y² +z²

A: The given limit is .The objective is to prove the limit does not exits.

Q: 2ry? does not exist. now that the limit lim (z,y)+(0,0) x² + y!

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Q: 5. *Let I = (-∞, 2) U (2, +∞) and consider the following real-valued functions defined on I. x²-4 1…

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Q: 6. Prove that if f RR is continuous with limx→∞ f (x) limx→∞ f (x) f(c) > L for some c E R, then f…

A: The answer provided below has been developed in a clear step by step manner.

Q: 5) Compute the following limit if it exists 2 (x+y)³² - (x-g) ty lim (x,y)=(0,0) (012)

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Q: 4. Consider the graph of function f (x): a) at x = 2 f(x) A) is continuous and differentiable B) is…

A: We are given a graph and we need to answer some points on the basis of the given graph:

Q: 3. Show that the limit xy lim (x,y)-(0,0) x² + y² doesn't exist. Show work!

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Q: 8. Suppose that f is a continuous function with f(x) > 0 for all real numbers x, and lim f(x) = 0 =…

A: Consider the provided question, Suppose that f is a continuous function with f (x)&gt;0 for all real…

Q: 10. Prove that the statement "If lim f(r) exists, then f is continuous at a" is false by using a…

A: See the solution below.

Q: Let Y = {−1, 1}. Then, (1) There is no continuous function f : X → Y , where Y is equipped with…

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Q: 5. If f (2) = 5 for the continuous function f (x), calculate the following limit value | F(t) dt lim…

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Q: 6. ) Express the limit as a derivative and evaluate. x2021 - 1 lim x - 1

A: we have to solve the limit problem