Given that for each a E (0, 1) lim t "(1 – t)"dt exists. Let, this limit be g(a). In addition, it is given that the function g(a) is differentiable on (0, 1). The value of g is

Q: If lim (æ) – f(a) x - a | exists, then f is differentiable at a. Select one: True False

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Q: Let f(x) be a function of one variable defined near x = a. Given a number M, set

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Q: Let f and g be real-valued functions defined on (b, ∞). Suppose that limx->∞f(x)=L and…

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Q: 15. Vlad and Karl discuss about the continuity of the given function. Vlad asserts that the function…

A: A function f(x) is said to be continuous at x=a iff limx→a-f(x)=limx→a+f(x)=f(a) A polynomial…

Q: Prove: For all integers a,b and c, if a|b and a|c, then a|(3b–2c)

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Q: Question: 1 a) Find and sketch the domain of the function f(x,y) = + Vx+1- log:(x² + y² – 4). b)…

A: Given function is fx,y=-11-x+x+1-log13x2+y2-4                   (i) To find the domain of fx,y Now,…

Q: State the Fundamental Theorem of Calculus (Part I and Part II). Hence or otherwise: (a) If u is a…

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Q: 5. Determine the infinite limit. a). f(x) = lim +1 x-5 2-t b). ƒ(t) = lim t→1 (t–1)2 x→5-

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Q: Let k ER\{0}. Give an example of a function f : R² → R² and c E R² such that lim || f(x) || = k but…

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Q: Let f be a function defined on an open interval I containing a. Prove that f'(a) exists and is equal…

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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: 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: calculate the limit of the function limT(æ → 0) [cosx/x2]]

A: To calculate the limit of the function.

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: In a model of the nervous system, the intensity of excitation of a nerve pathway is given by I = E[1…

A: Given  that: I=E1-e-a(S-h)E

Q: 17. 1+3n 0.

A: Alogrithm: Obtained P(n) and understand it's meaning. Prove that statement P(1) is true. Assume…

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

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

Q: 4. Let function g: R → R be defined by g(x) = |x|. g(0+h)-g(0) h B. The value of lim 9(0+ h) − g(0)…

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Q: Prove that if S(c +h) – f(c) lim h exists, then f is continuous at c.

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Q: 1. Consider the function given by if r € QU [0, 1] f(x) = 1- I 1 ° if r E Q°U [0, 1] Determine if…

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Q: ose F: R" → R² is of the form F(r) = (f(x), g(x)), rE R". Show that lim F(r) = (a, b) fand only if…

A: Introduction: A function f(x) is continuous at the point an if and only if the three conditions…

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

A: we have to solve the limit problem

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.