8. Suppose that f is a continuous function with f(x) > 0 for all real numbers x, and lim f(x) = 0 = lim f(x). Prove that there is some number y such that f(y) > f(x) for all r.

Q: For a function f, suppose lim f(x) does not exist because the left – hand and right – hand limit…

A: Given limx→c f(x) does not exist because the left-hand and right - hand limit have different values.

Q: Suppose that f(x, 0) = 3 for all x ≠ 0 and f(0, y) = 5 for all y ≠ 0. What can you conclude about…

A: The limit of the function, f(x,y) at a point (a,b) does not exist if two paths upon which the…

Q: Let f(x)={−2x+2  x≠1               {2   x=1. What is f(1)? If f(1) does not exist, type: 99999 What…

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Q: Let f(x)={5x+1  x≠4               {1       x=4. What is f(4)? If f(4) does not exist, type: 99999…

A: The given function is

Q: Let f : [1, ∞) → R be a continuous function such that lim|f(x)| = 0. For n ≥ 1, let gn : [1, ∞) →…

A:

Q: Suppose f(– 1) = 0 g( - 1) = 0 f'(– 1) = 5 g'(– 1) = 4 and all functions are continuous. lim…

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Q: 8) Suppose if x 20 f(r) = if x < 0. Identify a function g with domain R such that lim(f +g)(x)…

A:

Q: Consider the piecewise defined function x² + 2x – 1 if x 0 ° Which of the following statements is…

A: As we know that,  the function fx is continuous at a point  x=a if the left-hand limit is equal to…

Q: 2 Use the definition of continuity to show f (x) = is x+3 continuous at x=1. Whenever you find a…

A:

Q: 1. Prove the following limit. a) f(x)=x²+2x+6 where lim f (x)= 21 b) lim x-x+1=-5 x3 c) limx =8 x→2

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Q: Let f(x)={−2x+2   x≠1               {−5         x=1. What is f(1)? If f(1) does not exist, type:…

A: Given,

Q: Let Зх — 2. if x + 2, f (x) = 5, if x = 2. (a) Find lim→2 f(x) (b) Prove your answer to Part (a)…

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Q: Suppose that f: R→R is a continuous function, satisfying f(5)=8. What is the value of the limit lim…

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Q: If f is continuous from the right of x1 and lim f (x) exists, then lim f'(x) = f'_ (x1). O True…

A: it is true

Q: If f(x) is differentiable at X =c then lim f(x) must exist and f(c) must be defined

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Q: Consider the following function. 1 f(x): X + 9 Find the limit. lim Ax→ 0 f(x + Ax) - f(x) Ax

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Q: Select the three reasons for the non-existence of lim,>c f (x). lim, c f (x) = 5 x→c lim, »c- f (x)…

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Q: Let f(x) = 2x² + 2x + 3. According to the definition, f'(x) = lim h→0 (Enter the limit in reduced…

A: the soluton is following.

Q: Let f(x) V4x + 6. According to the definition, f'(x) = lim (Enter the limit in reduced form)

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Q: Suppose that f(x) 0 for all a in the interval (-1,1). If lim f(x) = L, then %3D L> 0. For those who…

A:

Q: 2. For the given function, f, determine if the function is discontinuous at a given point. State…

A: Given function is fx=xx. To find whether function is continuous at x=a, where a=0. Solution:   we…

Q: 1) Describe in your own words, how to determine the limit of f(x) as x approaches a given point and…

A: Hi! you have posted multiple questions. As per norms, we are answering the first question as nothing…

Q: Suppose that for a function f, f(3) is not defined. Also suppose that lim f(x) = 2 and lim f(x) = 2…

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Q: Prove that a function f is continuous at a limit point p iff the limit as x goes to p is f(p).

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Q: Suppose if x >0 f(x) = -1 if x < 0. Identify a function g with domain R such that lim (f + g)(x)…

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Q: 1. Show that for all f e L'(R) lim %3D

A: Put the delta = 1 in integral function

Q: 4. Do you agree with the following statement? When a function f is continuous at a number a, then it…

A:

Q: Let f(x) = 3x² – 5x + 1 f(x+h)-f(x) (a) Find f'(x) using the limit definition of derivative. That…

A:

Q: suppose f,g an d h are functions which g(x)<=f(x) <=h(x) ,for all x in an open interval containing a…

A: No. For the given continuous open interval domain, the limit of f(x) does not exist.

Q: 1 Let f(x) According to the definition, 7 4x f'(x) = lim h0 (Enter the limit in reduced form) %3D

A: Givenf(x)=14x-7

Q: 15. Suppose that the functionf:R Ris differentiable at xg. Prove that xf (xo)- xof (x) lim = f(xo) -…

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Q: If a function f is not defined at x = a then the limit lim f(x) as x approaches a never exists.…

A: To check: Whether the statement is true or false: Given: The statement is " If a function is not…

Q: 10. Prove that the statement "If lim f(r) exists, then f is continuous at a" is false by using a…

A: See the solution below.

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: Suppose that f(x) = 1 if x 0. Does lim,→0 f(x) exist? (Draw a picture.) O No, because the limits…

A: Given, limx→0-=1limx→0+=2 Hence, limx→0fx does not exist  since both the limits are not the same.…

Q: If a function f is not defined at x = c, then limf(x) does not exist. True False

A: Limit of a function at a point exist mean left hand side limit and right hand side limit at that…

Q: Show that the limit does not exist: (y² + x – 1)² - lim (x,y)→(1,0) (x – 1)2 + y4

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Q: - Define the function f:R → R by ƒ (x) = x +1 if x 0 and f (0) = 4. Show that lim,-0 f (x) = 1 and f…

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Q: Assume f is continuous on an interval containing zero anddifferentiable for all x not equal to  0.…

A: Consider the Darboux Theorem: Let I be an open interval, f:I→ℝ, a real- valued differential…

Q: Suppose that f(x) is a differentiable function (with a continuous derivative), such that f(0)=f'(0)…

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Q: Let f(x)={−4x+3   x≠4               {3       x=4. What is f(4)? If f(4) does not exist, type: 99999…

A:

Q: 1. Prove the following limits x 4y4 G) lim (x,y) (0,0) (x²+y4)3 = Does not exist

A: Solution is given below:

Q: A function f has a removable discontinuity at x = c if limf(x) exists by f is not continuous at x =…

A: Here, the given function is: f(x)=x-2|x|-2 So, function is not defined at x=2,-2

Q: show that L’Hôpital’s Rule is applicable to the limit as x →±∞and evaluate

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Q: If 2 - x2 &lt;=g(x)&lt;= 2 cos x for all x, find lim x--&gt;0 g(x).

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Q: Let F(x) = | 3e“ dt. Find lim F(x) if it exists. If the limit does not exist, explain why. x→0

A: see below the answer

Q: Let ƒ: S→R  be an uniformly continuous function. Finish the proof by showing that the limit lim_(x…

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Q: is continuous lim x f(x) = 0 = f is continuous in 0.

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Q: Suppose that lim, →o(f(x) – 9(x)) = 0 and h(x) is a function with domain R. a) Assume that |h(x) –…

A: We will answer the first question as we only answer one question at a time. Please resubmit the…

Q: 3x – 1, if x 8 | Let f(x) = { 5 x+9 Show that f(x) has a jump discontinuity at x = 8 by calculating…

A:

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.