Let f(x) =, where Lx] is the greatest integer function. Show that for x + 0, 69. Then use the Squeeze Theorem to prove that H- lim x x+0 Hint: Treat the one-sided limits separately.

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Q: Determine the following limits analytically.

A: Given: L=limx→-1x2+8-5-4xx3+1 For evaluating given limit, we substitute x=-1+h, h→0 So,…

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A: Since you have asked multiple question, we will solve the first question for you as per our guide…

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Q: Let f(x) = /3r – 10. According to the definition, f'(r) = lim h0 (Enter the limit in reduced form)

A: f(x)=3x-10f'(x)=limh→0f(x+h)-f(x)h

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A: Definition of limits helps to find the limit of the function. A value is said to be the limit of…

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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: Let f(x) = 4x² + 6x + 2. According to the definition, f'(x) lim h→0 (Enter the limit in reduced…

A: Given f(x)=4x2+6x+2

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

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Q: Let f(x) =, where [x] is the greatest integer function. Show that for x 0, 1 - 1 < Then use the…

A: Definition used -                        Squeeze theorem -                          If…

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

A: Recall the formal definition of the limit to prove the result as follows: limx→3fx=limx→3x2-2x-32x-6…

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A: Follow next step

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Q: Use the Squeeze Theorem to evaluate the limit lim f(x), if 2x 1< f(r) < 2 on [-1, 3].

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Q: When using the definition of limit to prove that L is the limit of f(x) as x approaches c, you find…

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Q: Prove the following limit, where the Squeeze Theorem is used: lim, 0(xtan-1(x)) = 0

A: Let us try to understand concept of Squeeze theorem-

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Q: Use the Squeeze Theorem to find the following limit when c = 0, 2 – x² s f(x) < 2 + x². lim f(x)

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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 formal definition of limit to prove that the function is continuous at c. h(x) = sqrt(2x -…

A: To show that the function is continuous at x=2 we will find the left and right-hand limit at x=2.

Q: Use the Squeeze Theorem to find the following limit when b – |x – a| < f(x) <b + |x – al. lim f (x)

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Q: Express the limit as a definite integral. [Hint: Consider f(x) = x2.] lim i = 1 n' xp 1.

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Q: Use the Squeeze Theorem to prove that lim væ[1+ (sin(27/x))²] = 0.

A: "Since you have asked multiple questions in a single request, we will be answering only the 1st…

Q: Evaluate using the Squeeze Theorem.

A: Given: limx→0 x sin 1x2  To evaluate: The limit using Squeeze theorem.

Q: Use the Squeeze Theorem to find the following limit when b - |x - al s f(x) sb + x - al. lim f (2)

A: 1.b-x-a≤f(x)≤b+x-alimx→af(x) Note:     Multiple questions are given, we are answer with only one…

Q: a)Express the limit as a definite integral. Σ (3c-5)Ax, xe[0,1] lim 2 b) and then evaluate it.

A: lim∆→0∑i=1n3c2i-5∆xi  x∈[0,1]∑ =∫∆xi=dx∫01(3x2-5)dxThis is definite integral

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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: Express the limit as a definite integral. lim 3 Xk, where P is a partition of [2, 3] k=1.k

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Q: use the formal definition of limit to prove that the function is continuous at c.  ƒ(x) = x2 - 7, c…

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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: Evaluate the following limits, if they exist. (Do not use L'Hospital rule) Vh? + 15 – Vh + 15 Vh +3…

A: follow next step

Q: If x Vx sf(x)s -x+ 1for all positive x-values, then what is lim f(x)?

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Q: if r =-1 A if z + -1 2. Is f(r) = continuous at r =-1? (Show your work using limits.)

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Q: Compute the limit of the function at −1 AND determine if the function is continous at 1. f (x) = {…

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