Question 2 Is the given statement true or false? Justify your answer. 'If f is continuous, decreasing function on [1, o0) and lim f(x) = 0, then f(x) dx is convergent." %3D

Q: Ql) check the continuity of fX) at X=-1, 0,1.23 -1<x<1 x-4 2x-1, 3x-2 2, x=3 f(x) = 1sxs2 25x53 Q2)…

A: 1)

Q: (i) Consider the function f: R\{-2,3} → R defined by 2x³4x²5x - 3 x²-x-6 f(x)= - (a) Prove, using…

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Q: 1. Express the limit as a definite integral of a function on certain interval and compute the…

A: using the particular form  1n∑r=1r=nfrn=∫01fxdx   and to find the  the derivatiove of function under…

Q: 2. cuts the curve at Q, such that Let ƒ be a continuous and differentiable Tunction in (x,, x.). If…

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Q: 2. If f is continuous on [0, 1] such that x* f(x) dx=0, for n=0, 1, 2, 3,... then show that f(x)=0…

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Q: 5. If g is continuous on an interval [a, b] and has an extremum at a number c € (a, b), then the…

A: I give two examples for this statement.

Q: 11. Suppose that f is uniformly continuous on [0, M) for all M > 0 and lim, f(x) true if we do not…

A: The complete solutions are given below

Q: Is the proof correct? Either state that it is, or circle the first error and explain what is…

A: The given proof is correct.

Q: Let y = f(x) be a continuous function. If ƒ has a local maximum or minimum value at x = a, then the…

A: Let y=f(x) be a continuous function. If f has a local maximum or minimum at x=a,then the limit…

Q: Evaluate the one sided limits of K at x=L algebraically using ly Usina a plecewise formula for…

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Q: Əsoddns f(x) is a continuous positive function, and n 2 2k Sn Es(1+ k=1 a Riemann Sum (with…

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Q: If f(5)-0, then which of following must be true? fis continuous at x-5 f has a horizontal tangent…

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Q: Use the definition of continuity and limit value (the epsilon-ô definition") to show the following:…

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Q: The funciton f in the figure satisfies the lim f(x)=5 as x approaches 4.  Determine the maximum…

A: (A) From the inequality |f(x)-5|&lt;1 , we\'ll find range of f(x) first.

Q: 1. Discuss the continuity of f(x) with graph shown below by answering the following questions.

A: Graph of the function is given, we have to the limit at x=0

Q: Suppost The following poroperties: f(x) = Trx, lim f)=2 that a function satifies %3D Find f(0), What…

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Q: f (х, у) %3D х4 + 4y2/x2 + 2y2 f function is defined as. Does f function have a limit at point…

A: We will evaluate the limits using two paths and see if they are same. If not same, the limit does…

Q: • Give an example of a continuous function f(x) on (a, b) such that f(x) has no maximum on (a, b).…

A: We have to give an example of a function f(x) that is not bounded on [a, b].

Q: Can i get step by step help please

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Q: If r(5)=0, then which of following must be true? Olimf(x)=0 x+5 f(x)-f(5) -o lim x-5 X-5 fis…

A: Here we have given that  f'(5) = 0 We have to find which option is correct.

Q: If f is defined on [0, +inf] and Lebesgue integreable, uniformly continuous function, then, lim_(x…

A: Given If f is defined on [0, +inf] and Lebesgue integrable, uniformly continuous function, then,…

Q: True or false? The limit of 1/x as x approaches zero is zero.

A: The given statement is false. Use the property of limits to solve the limit.…

Q: If lim | f(x)dx does not exist, b00 a then | f(x)dx diverges. a Select one: False True

A: True  Improper integral said to be convergent if the limit is finite and that limit is the value of…

Q: 7. Let f(2) = f(x, y) = + i , = 0. when z +0 and let f(0) %3D 5xº+y² (a) Show that lim f(z) = f(0) =…

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Q: A function is defined by: In( x- 1) f (x) =- 1-x a) Using L'Hopital's rule, show that: lim f (x) =0,…

A: Critical points

Q: 3. True/False: if the limit of a function exists at x=c, then f(c) exists.

A: The given statement is

Q: Use the definition of continuity and limit value (the epsilon-8 definition") to show the following:…

A: Given,f:R →Rf is continuous in x = 0

Q: (a) Is f continuous at the point (0,0)? Justify your answer. (b) Does the equality lim lim f(x, y) =…

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Q: The continuous extension of (sin x)^x [0,π] a) Graph f(x)= (sin x)^x on the interval 0 ≤ x ≤ π.…

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Q: Suppose that f has continuous derivative and is twice differentiable in R. Suppose that f(a) = f(b)…

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Q: Let fn: R --> R be defined by : fn(x)= x/(1+nx2), For all n >= 1. a) Show that {fn} converges…

A: step:-1 Theorem:- Let fn be a sequence of functions on I such that for each n∈ℕ, f'n(x) exists for…

Q: True or false 1. If the left- and right-hand limits of f(x) as x→ c exists and are equal,Then f(x)…

A: If f(x) be a function on a subset of real number and c be point in domain of f then f is continuous…

Q: Show using sequences of functions that 2 lim e dx = 0.

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Q: 2. Consider a function f(x) = cos a² + (5)². %3D (1) Show that f is continuous on R. (2) se the fact…

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Q: (a) What is wrong with the following use of l'Hopital's rule? x° + x – lim +1 х2 — Зх +2 3x2 +1 =…

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Q: Does the existence and value of the limit of a function ƒ(x) as x approaches c ever depend on what…

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Q: Does the existence and value of the limit of a function f(x) as x approaches a ever depend on what…

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Q: H(3)=6 and the limit as x approaches 3 of H(x) is 6. Mustache be continuous at (3,6)? State why.

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Q: 3. Evaluate (3x + 1)dx by using the limit definition with the right end points. (Don't use the…

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Q: True or false: Let f(x) be a continuous function. Suppose that f(x)dx converges for some real number…

A: We have used some properties of integration.

Q: 25. Let f(x) be a continuous function. Express + ()] lim п—оо as a definite integral.

A: Definite integral in the form of the limit of the Riemann sum:   Let f(x) is a continuous function…

Q: ). Determine the interval(s) on which f(x) =4x−12x−1is continuous. b). Find lim x→0…

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Q: TRUE or FALSE. If f is undefined at x = c, then the limit of f (x) as x approaches c does not exist.

A: Given statement If f is undefined at x=c, then the limit of f(x) as x approaches c does not exist.

Q: II. Given: ‘sinh r − 1 if x 2 x-2 Determine if f is continuous or discontinuous at x = 0 and at x =…

A: Types of discontinuous of the pice wise continuous function

Q: F(x) Let F(r) = | 3e" dt. Find lim if it exists. If the limit does not exist, explain why.

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Q: Determine the limit of the function, f(x), by squeezing it between an upper bound, h(x), and a lower…

A: Squeeze theorem states that let I be an interval having the limit point b. Let g, f, and h be…

Q: Σ 1 is continuous function in R. Prove f(x) = 4 n=1 n' +x*

A: The given function is                                     f(x)=∑n=1∞1n4+x4 To show that the above…

Q: Use the definition of continuity and the properties of limits to show that the function is…

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Q: If f is positive-valued continuous decreasing function and lim - S(x) = 0 then J F(x)dx diverges…

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Q: You may assume that the exponential and cosine functions are continuous and may freely use…

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.