Given that lim f (x) = L and lim g(x) = M andf (x) < g(x) for all x in an open interval containing c (except possibly c itself), prove that L< M.

Q: Use the Squeeze Theorem to evaluate the limit lim x→4 f(x)  if 8x − 16 ≤ f(x) ≤ x^2   on[2,6].

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Q: 1. Evaluate the following limits (if it exists) (a) lim X3 Vx+6-x (b) lim ズ→1 Vx-1+x2-x ズー3 xー1 2x…

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Q: Consider the limit limx→0 1 x 2 = ∞ and let M = 1000. Find the largest value of δ > 0 such that if x…

A: The definition of limit: Let fx be a function and limx→cf(x)=∞. Then, for every M&gt;0, there exists…

Q: II. Prove that for each of the following lim (x,y)→(0,0) f(x, y) does not exist. ,2,,2 4. f(x,y) x4…

A: limx, y→0, 0x2y2x4+y4

Q: Suppose that c is a real number and that lim f(x) and lim g(x) both exist. Give the 12 limit…

A: Use the given limit conditions and write the theorems. 1…

Q: 2. Use the ɛ-d definition of the limit of a function to prove that lim = 3. I-»3 T – 2

A: Let  I  be an open interval containing  c , and let  f  be a function defined on  I , except…

Q: Find the limit. lim x→0+ (- 7lm a)" In x

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Q: If f(x) s g(x) for all x in some open interval containing c, except possibly x = c itself, and the…

A: Here we have to determine the following statements are true or false.

Q: Prove that if f(x) > 0, lim f(x) = 0, and 6. х а lim g(x) = ∞, then lim f(x)s(x) = 0. х а ха

A: To prove:  Formula used:

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: Evaluate the following limits analytically, if it exists. (a) Vx+14 – 1 lim x→-10 2x (b) 5x +7x lim…

A: Given problem:-

Q: Prove, using the e-8 definition of limit, that: (a) (b) lim ²+1=2 24-1

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Q: 3. Evaluate the limit of lim(x² + x + Vx + 1) x→3

A: A limit of a function f(x) is defined as a value, where the function reaches as the limit reaches…

Q: If f is an odd function and if lim f(x) exists Prove that this limit must be zero.

A: if limit is exist then right hand limit and left hand limit must be equal

Q: Find the limit.   Lim  x→1+      x1/1-x

A: Given function is By property of logarithm

Q: Is P(x) continuous at x= 3? O No, because lim P(x)+ P(3). x-3 O No, because lim P(x) does not exist.…

A: Given,          x=3 To determine if P(x) is continuous at x=3.

Q: Find all values of c such that the limit exists.

A: Our Aim is to find the value of c from the given expression such thatthe limit…

Q: 4-IS. 2-x s (x)< 2Cos(x) for all x then find: lim f(x)

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Q: draw the graph of a function on [0, 5] with the given properties.

A: Removable discontinuity at any point (a) of domain of function (say f(x)) is define as Graphically…

Q: find all values of c such that the limit exists

A: Given

Q: Given a function f:R → R with f(x) = x - 1, for x rational and f(x) = 5 - x, for x irrational Prove…

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

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) Evaluate the indicated limits, if exist (x³ + 1) -x (а) lim r+0 V9 + x² (b) lim In I→1+ (c) lim…

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Q: Given that lim f(x) = - 8 and lim g(x)= 2, find the following limit. X→3 x-3 lim [8f(x) + g(x)] lim…

A: topic - limits

Q: If f(x) > 0 for all æ and A = lim f(x) exists, then A > 0.

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Q: Prove that lim ƒ(x) = L if and only if lim ƒ(h + c) = L.                   X-&gt;c…

A: Given : The statement,  limx→cf(x) = L if and only if limh→0f(h+c) = L

Q: If if x is rational f(æ) = 1 0 if x is irrational prove that lim f(x) = 0.

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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: The value of k such that x2 lim kx + 7 x→1 x – 1 exists is

A: Given: The limit expression limx→1x2-kx+7x-1, To determine: The value of k such that the given limit…

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: Evaluate the limit. lim 3 x-1 O-1 O1 O3 Does not exist

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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: evaluate the limit.

A:

Q: 2. Assume that for x # c (in some open interval containing c), I(x) < f(x) < u (x) and that lim 1(x)…

A: lx ≤fx≤ux and that limx→c lx =limx→c ux =L Now to show that limx→c fx  exists and limx→c fx =L  Here…

Q: Determine the value for c so that lim f(x) exists. x-3 f(x) = 5**, *+c, for x 3

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Q: Show that the lim Z→0 |Z| does not exist.

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Q: B. If k is any real number, prove or disprove that sinkx 1. lim x-0 kx = 1.

A: We will prove this by L'Hopital's rule

Q: With regard to the limit lim x² = 25, Find a 8 such that |x² – 25| < 10¬3 if 0 < |x – 5| < 8.

A: We are given limx--&gt;5  X^2 = 25 We have to find delta , and we can find it by applying definition…

Q: Prove that if lim-→0 f (x) exists, then there exist & > 0 and some M > 0 such that |f(x) < M for 0 <…

A: We have to proof the given statement.

Q: It was determined that the lim f(x) = 0 and f(2) = 0. Can it be concluded, definitively, that f'(2)…

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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: 44. Prove that if lim f(x)= L #0 and lim g(x)= 0, then the limit lim 8(x) does not exist.

A: Given: limx→cf(x)=L≠0 and limx→cg(x)=0 To Show- limx→cf(x)g(x) doesnot exist. Theorem- if limx→au(x)…

Q: Find the limit. x* – 1 - lim x→0+ In (x) +x – 1 п lim ( 1+ n

A: These question is based on Limits.

Q: Given lim f(x)=L, Im fx)=L, and L, =L, 1. lim f(x)=L, II. lim f(x)=L, III. lim f(x) does not exist

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Q: Prove, using the formal definition, that lim,→1- V1– x = 0.

A: Given, limx→1-1-x=0

Q: Find the limit lim x3 e-x2 X→00

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Q: Find Limit Using Squeeze Theorem: 3. If 2x s g(x) s x* – x² + 2 for all x, find lim g(x).

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Q: 6. Explain why the limits lim f(x+h) – f(x) and lim f(2) – f(x) h Z-X are each initially in the form…

A: Given that the limits are limh→0fx+h-fxh and limz→xfz-fxz-x.

Q: Prove rigorously that lim (x2 – x) = 6. x→3

A: We have to prove  limx→3x2-x=6 Now let  L=limx→3x2-x