Suppose E is a subset of X, where X is a metric space, p is a limit point of E, f and g are complex functions on E and the limit as x approaches p of f(x) is  A and the limit as x apporaches p of g(x) is B. Prove the limit as x approaches p of (f/g)(x)=A/B if B does not equal 0

Q: The integer n = 12700140x3243243 is missing the digit x. a) If n congruent to  3 (mod 9), then what…

A: According to the given information, Consider the given integer n:

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…

Q: Use the &,&-definition of a limit to prove the following limit. lim (az + b) = azo+b, for complex…

A:

Q: cos 2x–1 lim COS х— 1 cos x-1

A: limx→0cos 2x-1cosx-1

Q: 2. Prove mx+b=mc+b   using the definition of a limit of a function where fx=mx+b is a function where…

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

Q: Let K' be and K' be to divisions of [a,b]. Show that  L(f,K') ≤ U(f,K'')

A: We need to prove the following : If f is a bounded function on [a, b], and if P and Q are partitions…

Q: Evaluate the limit V10+102 lim HI00 11+10r

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Q: S* 56sin(t")dt Evaluate the limit lim .8

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Q: 1) Evaluate the limit in each part if it exists Vx+2 -3 a) lim x-7

A: To evaluatelimx→7x+2-3x-7

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: Prove that lim (2a2 + 3r + 6) = 5 using the e, 8 definition of a limit. %3D a -1

A:

Q: Let g : R → R be a bounded function and define f : R\{0} →R by 1 f(x) = g(x)+ Using only the…

A: It is given that ;                f : R \ 0 →R  defined by     fx = gx + 1x2 Where,   gx  is a…

Q: Consider the function G: R²\{0} → R defined by 12,8 (x10 + x10)² G(x): = Find an expression for the…

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Q: 6 Use the e - d definition of a limit to prove e lim 1+2 r2 + 2 1.

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Q: Evaluate sin 0 – 1 lim 0 - T/2 In sin 0

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Q: Suppose that you have $11,000 in a rather risky investment recommended by your financial advisor.…

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Q: Find the limit of f (x) = √(14 + √x), defined on X = [1, ∞)

A: The given function is: f(x)=14+x  is defined on [1,∞).

Q: 1-cos 2x lim 4x

A:

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: find all values of c such that the limit exists

A: Given

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: х2—1 lim х-1+ In(x)

A:

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: Which of the following functions are homo- morphisms? (a) g: R→ M₂(R) given by g(a) = (a 6). (b) h:…

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Q: lim f(x) = f(0). X-0

A: We have to prove that  limx→0+f(x)=f0

Q: Suppose that f(x) = u(x, y) + iv(x, y) and zo = xo + iyo. Prove that lim f(2)=uo + ivo 2-20 if and…

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Q: Olim x→1 1 x+1 x 1 - 1|2

A: Explaination of the answer is as follows

Q: ind the limit as x approaches two of (2x + x – 10)/(2 – x)

A: