Draw the graph of an example of a function, f, (ONE function) satisfying ALL the following properties. The function f is continuous everywhere except: a) f has a removable discontinuity at t, b) it has a jump discontinuity at 2t, but it is right-continuous there, c) it has an infinite discontinuity at 3t. d) And y = d is a horizontal asymptote for f but lim,→+∞ = +∞.

Q: use the formal definition of limit to prove that the function is continuous at c. g(x) = 1/(2x), c =…

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Q: Prove that ƒ is continuous at c if and only if

A: We want prove that f is continuous at c if and only if limh→0f(c + h) - f(c)  that is limh→0f(c + h)…

Q: Assume whether the function graphed in the accompanying figure is continuous on [-1,3]. y = k(x) 2 3…

A: for being a contrinuous function this condition should be satisfiedlimx→a-f(x)=limx→a+f(x)=f(a)from…

Q: (a) Consider a function f: R → R whose graph Gf is a closed subset of R?. Use the sequential…

A: For (a), We are given a function f : ℝ→ℝ whose graph Gf is a closed subset of ℝ2.  If f is also…

Q: Suppose that f has a one-sided limit at each point of [a,b]. Prove that f is bounded on [a,b]

A: According to the given information, let f be one sided limit at each point of [a,b]. It is required…

Q: Sketch the graph of a continuous function f that satisfies all the stated conditions: f(0)=4,…

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Q: Let f(x) = – and let g be the function defined by (x – 3)(x – 5) if x 3 (a) Evaluate lim(ƒ o g)(x).…

A: Ans:-  a) - infinity  b) infinite discontinuity  Explanations:- In part (b) as LHL is not equal to…

Q: Let f be a real function, defined everywhere on R. Prove that if lim f(x) = L, then also lim f(-x) =…

A: Here we will use ε-δ  definition of limit Let f(x) be a function defined on an open interval around…

Q: Suppose that f is differentiable on some open interval containing c, and (c,f(c)) is a point of…

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Q: Prove that ƒ is continuous at c if and only if lim hS0 ƒ(c + h) = ƒ(c).

A: Prove that ƒ is continuous at c if and only if limh→0fc+h=fc.

Q: Given the function, g(x) x 2 a. Determine the possible points of discontinuity. b. Discuss the…

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Q: 2. ASSUME f(x) IS CONTINUOUS AND DIFFERENTIABLE ON AN OPEN INTERVAL AS GRAPHED. WHICH OF THESE…

A: Since you have posted multiple question in this question so I have solved first question as per our…

Q: use the formal definition of limit to prove that the function has a continuous extension to the…

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Q: use the formal definition of limit to prove thatthe function has a continuous extension to the given…

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Q: Given the function, g(x) 3 x 2 a. Determine the possible points of discontinuity. b. Discuss the…

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Q: use the formal definition of limit to prove that the function has a continuous extension to the…

A: Given function is fx=x2-1x+1 and point is x=-1

Q: (-3,0) 3 -2 4- 3+ 2. -1 (0, -0.5) -1,-2) (-1,-3) -3- (1,0) 2 y = f'(x) 3 (2,-0.5) (3,-2) (4, -3) I

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Q: Use the IVT to show that if f is continuous and one-to-one on an interval [a, b], then f is either…

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Q: Suppose that f has a one-sided limit at each point of fa, b. Prove that f is bounde

A: In mathematics, a function f defined on some set X with real or complex values is called bounded if…

Q: Given the function, g(x) 3 x 2 a. Determine the possible points of discontinuity. b. Discuss the…

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Q: Determine the one-sided limits at c = 2 and c = 4 of the function f in Figure. What are the vertical…

A: Consider the given graph: Determine the one side limits at c=2 and c=4.

Q: From the graph of f, state each x-value at which f is discontinuous. For each x-value, determine…

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Q: Each of the following statements is false. For each statement, sketchthe graph of a function that…

A: The first statement (a) is, If limx→af(x) exists, then f is continuous at x=a. The statement is…

Q: Give an example of functions f and g such that f (g(x)) is continuous but g has at least one…

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Q: Suppose f and g are real-valued functions defined on all of R, f is measurable, and g continuous. Is…

A: Given, that f and g are real valued function on ℝ and f is measurable and g is continuous function.…

Q: Suppose that ƒ : [a, b] → R is a continuous and one-to-one function. Prove that ƒ is strictly…

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Q: B. Show directly from the definition (without using Arithmetic of Limits or similar results) that…

A: Let's Solve,

Q: Give an example of functions f and g and a point x = a such that the limit of f as x approaches a…

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

A: This question can be solved using the concept of continuity of a function at a given point.

Q: A function f is said to have a removable discontinuity at x = c if limf (x) exists but f is not…

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Q: 34. Draw a graph that is continuous for all x, with no cor- ners, but not differentiable at x = -1…

A: Continuity for all x with no corners but not differentiable at x=-1 and x=2

Q: Let f(a) = a 8-Va Calculate lim f(a) by first finding a continuous function which is equal to f…

A: Given function is fa=64-a8-a Calculate lima→64 fa by first finding a continuous function

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

A: Givenpv=52v2+2,a=1

Q: Use the table of values of f to estimate the limit. 4) Let f(x) = x2 + 8x – 2, find lim f(x). х-2 4)…

A: topic - limits

Q: Let g(x) be the following piecewise-defined function: 9 – x2 if x + -3 g(x) = 2(x+3) B2 if r = -3…

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Q: use the definition of differentiability f'(x) = lim- h-0 f(x+h)-f(x) h to For f(x) = %3D 6. (x-1)…

A: fx=x2x-1 Now,…

Q: This function: 2. 2 0 4 2 -4 O has a removable discontinuity is continuous has a jump discontinuity…

A: Given: To find the behavior of continuous function.

Q: Use the definition of continuity and limit laws to prove the following statement from Theorem 4,…

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Q: If ƒ is a continuous function on a compact set, show that either ƒ has a zero or ƒ is bounded away…

A: Given, if f is a continuous function on a compact set. To prove that either f has a zero or f is…

Q: Given the graph of f(x) shown below, which condition in the definition of continuity fails first at…

A: Continuous function=> left-hand limit and right-hand limit for a function x = a exists and  they…

Q: Let f(x) = and let g be the function defined by (x – 3)(x – 5) if x 3 valuate lim (f o g)(x). fog…

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Q: Given that lim f (x) = L and lim g(x) = M and f(x) < g(x) for all x in an open interval containing c…

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Q: Let f (x) = (x-x-2) х-2 1. Show that f (x) is discontinuous at x = 2. 2. Determine whether or not x…

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Q: Let f : [0, 00) → R be a monotone function. Prove that lim f(x) = L for some number L if and only if…

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Q: From the 8, e definition of Continuity, prove that the following function is con- tinuous at the…

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Q: 8. lim f(x) = 3, lim f(x) = ∞, lim f(x) = –o, f is odd x2- X2+

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Q: Sketch the graph of a function f that is continuous except for the stated discontinuity. The…

A: Removable and jump discontinuity

Q: Use the definition of "limit of f at c" to prove the limit of lim, 3(x² – 5x + 1) = -5

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Q: From the 8, e definition of Continuity, prove that the following function is con- tinuous at the…

A: Here we use definition of continuity.

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.