Let f be defined by:                1,   x rational f(x)={                0,   x irrational ;prove that for no number c does limx->c f(x) exist

Q: 4 lim X-0 x2 2 1-cosx

A: Given- limx→04x2-21-cosx To Find- The value of the limit.

Q: Find Limit Using Squeeze Theorem 2. If 2x – 2 < f (x) < x² – 4x + 3 for x 2 0, find lim f (x). - x→1

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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: Is there a number a such that Зx? + ах + а + 3 lim x? +x – 2 1--2 exists? If so, find the value of a…

A: We have given;

Q: Let a be a nonzero constant. Prove that if lim f(x) = L, then lim f(ax) = L. Show by means of an…

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Q: Let n be a positive integer. For which n are the two infinite one-sided limits limx→0±1/xn^n equal?

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Q: 3 Use the Squeeze Theorem to evaluate lim f(x), where f(x)=x²cos (-). x A B C D - 0 0 1 8 -0-x

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Q: Find the value of a, b, andc such that the limit a + bx + cx - cos(2x) lim x4 exists. a = (Enter…

A: →When limit is of the form 00 or ∞∞ then L'Hospital rule is used.     In this rule, take the…

Q: Compute the limits below, and explain why they are not indeterminate limits. a) lim as x approaches…

A: Explanation of the answer is as follows

Q: Prove that lim r(x)= where m = 0, 1, 2, 3. .. II

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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: By using Sandwich Theorem, prove that lim sin x x-0 r

A: We have to prove that limx→0sin xx = 1 using sandwich theorem. Sandwich theorem states that- Suppose…

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: Q: Prove ha中 Lim x" an %3D

A: we have to prove that the following

Q: Enter a whole or decimal number, or DNE for does not exist or undefined. lx-이 4 + | lim x →0-

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Q: Let f : (a, b) → R be differentiable and let c ∈ (a, b). Suppose that limx→c f‘(x) exists and is…

A: handwriting solution

Q: Prove that if lim, „a g(x) = 0 and lim, „«f(x) exists and is not 0, then %3D f(x) lim does not exist…

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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: Prove that lim |x|sin(1/x) = 0.

A: We need to prove that limit of given function is zero.

Q: Prove that lim(x,y)→(1,2) (x2 + 2y) = 5 by using the precise definition

A: ε-δ definition: Limit of function f(x,y) at a point (a,b) is 'L' if for every ε&gt;0 there exist…

Q: Prove that lim r(x) = 0 where m = 0, 1, 2, 3. ...

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Q: Suppose that f is decreasing on [a,b]. Prove that that b. lim Rpf) exists and equals to Ilp||+0

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

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Q: 2. Let r > 1. Prove that lim r" = 00,

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Q: Let if r is rational, f(x)= 0 if x is irrational. Prove that limx→a f(x) does not exist for any a €…

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Q: Prove that if lim x(t) = X and lim y(t) = Y, then lim(x(t), y(t)) = (X, Y). t-a t-a t-a

A: Now, xt, yt=∫cdxtytdt Given: limt→axt=X   , limt→ayt=Y

Q: 1. Prove the following limit. a) f(x)=x²+2x+6 where lim, 3 f (x)=21 x→3 b) limx? -x+1=-5 c) lim x→2…

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Q: Let f(x) = 8x + 3. (a) Prove that lim f(x) = 27 using the formal definition of the limit. 3 (b) Find…

A: Givenf(x)=8x+3To prove limx→3f(x)Subdivision b is not defined properly

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: エ→2ー Prove that lim sin(1/x) does not exist. Lot II

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Q: 2х — 5 lim 4x

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Q: Using the squeeze theorem, find the value of lim,0 x² sin=) O 1 O -1 O does not exist

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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: Prove rigorously that lim sin 1/x does not exist.                                    x → 0

A: Let the given limit be y - y = limx→0 sin1xy = limx→0 sin1x1x×1xy = limx→0 sin1x1x×limx→0 1xy =…

Q: Suppose that f is decreasing on [a,b] Prove that that lim Rpf) exists and equals to

A: let ∈&gt;0 be any real number  let P={a=t0&lt;t1&lt;t2&lt;......&lt;tn=b} be a partition of [a,b]…

Q: 30. Suppose that f, f', and f" are positive on R. Prove that lim f(x) = 00.

A: Let f, f' and f'' are positive on R. We need to prove that limx→∞fx=∞ Consider, the interval [a,∞).…

Q: Evaluate using the Squeeze Theorem.

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

Q: Let zo = 0. Discuss if the limit of function f(2) = exists at zo- • If yes, then lim f(z) = z+z0 •…

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

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Q: Use the Squeeze Theorem to show that lim x* cos = 0. If If(x) < B| for all x in an interval…

A: Squeeze (Sandwich)Theorem The Sandwich Theorem, also known as the squeeze theorem, is used to find…

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: 4 Using the e – 8 definition of limit, prove that lim x→3 3x 1 1 2

A: To prove the limit of given function using the epsilon delta definition of limit which states that…

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 E C(0, 1). Suppose that lim f(x) and lim f(x) exist (and are finite).

A: Given that f is continuous on (0,1)

Q: 4. Use the definition of a limit to prove that lim z = a* for any real number a.

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Q: evaluate the limit.

A: We want to evaluate the following limit.

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: 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: Prove that lim n 00 a" f (x) dx = 0, for any f E R[0, 1].

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