Exercise 2. Let f: (1, 00) → R and g : (0, 1) → R be such that g(x) = f(1/x). Prove that lim f(x) = L if and only if lim g(r) = L.

Q: (5) Let ƒ : (a, ∞) → R be such that lim xf(x) = L, L € R. Prove that lim f(x) = 0. X48 x→∞

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A: Given : (a, b) is on the boundary of the domain of f

Q: 3. Describe the interval(s) on which the function is continuous. c) S(x)= x-1 2, x= 1

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Q: If f(x+ y)= f(x)f (y) for all x, y eR and f (x)=1+g(x)G(x), where lim g (x)=0 and lim G (x)…

A: The problem has been solved in detail in the next step.

Q: Let E ⊂ R where a, b ∈ E, c ∈ R, a < c < b, with (a, c) ∪ (c, b) ⊂ E. Let f a real-valued function…

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Q: Find formulas for f and g such that limx→∞ f (x)/g(x) is an ∞/∞ form that has a limit of ∞.

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Q: If f is an odd function and if lim f(x) exists Prove that this limit must be zero.

A: if limit is exist then right hand limit and left hand limit must be equal

Q: 4. Let f RR and let cЄ R. Show that limf(x) = L if and only if lim f(x + c) = L. x-e x-0 PLEAEhelp…

A: The objective of this question is to prove that the limit of a function f(x) as x approaches a…

Q: 7 - Let F(t) = cosnti" + sinntj* + 6tk* G(t)* = 2ti + t³j* %3D %3D Find a) lim[F.G*](t) t-3 b) lim[F…

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Q: lim -1 x→0+

A: Separate the limit.

Q: Evaluate the limit lim f(t) where ƒ(t) = = lim f(t) = i 00-11 00-11 (9+17) 2 for t = 0.

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Q: Prove that lim r(x) = 0 where m = 0, 1, 2, 3. ...

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Q: x→4 5. Sketch a function f such that: lim f(x), lim f(x), and f(4) all exist, but no two are equal.…

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Q: Let a, b R and g: (a, b) → R be a function. Suppose that f: (a, b) → R is a bounded function.…

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Q: lim (x,y)-(-3,4) (x³ +3x²y² - 5y³ +1)

A: Here we have given an expression with limit and we have to solve it. We have lim (x,y)→(-3,4) (x3 +…

Q: Prove that lim [x + ¡(2x + y)] = 1 + i using the definition of limits. z+1+i

A: Using the definition of limits we have to prove that: limz→1+ix+i2x+y=1+i.

Q: Let f be entire and suppose that lim 80+z f(z) Z = 0. Show that f is constant.

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Q: Let F = {f ∈ C(R) : lim|x|→∞f(x) = 0 }. Show that F is closed in C(R

A: please  see the next step for solution

Q: Consider the funetion 1) Find the domain of glx) 2) Find lim g(x) and limg(x). 3) Find oblique…

A: Solution: Given data ,  g(x)= 2√(x2+3x)

Q: 2) Evaluate Lim X + 2² (X, Y, Z) (0, 0, 0) + Z 1 or show that it exists NOT.

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Q: x² – y + 1 f (x, y) x² – y? + 1 -

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Q: x² - 1 if x 2 x→2* а) 4 b) – 3 с) 3 d) – 4

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Q: Calculate lim f(x) or determine that the limit does not exist. x-12+ 60 - x x 12 X- 12

A: Follow the steps.