Suppose f and g are functions with domain R. If both f and g are even but f+g is odd, then prove that g(x) = −f(x) for all x ∈ R. b) Suppose neither limx→0 f(x) nor limx→0 g(x) exists. Show that limx→0 [f(x)g(x)] may exist.

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 g(x)<h(x)forx∈(0,1)∪(1,2) both limx→1g(x)andlimx→1h(x) exist. Must limx→1g(x)<limx→1h(x)?…

A: Given: gx&lt;hx for x∈0,1∪1,2 and both limx→1gx and limx→1hx exist, To determine:…

Q: Prove, using the e-d definition of limit, that: (a) lim x² + 1 = 2 24-1

A: As per our guidelines, we can answer only one question in a single request. Kindly re-post the other…

Q: 1. Use the definition to show that x² - y² lim (x,y)→(0,0) x² - y² = 0.

A:

Q: Use the Squeeze Theorem to find limx→c f(x).

A: Given that 4-x2≤f(x)≤4+x2 To find the limit of the function using the Squeeze theorem when x→c=0.

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

A:

Q: Let g(x) = (x2 - 2)/(x - √2). Make a table of the values of g at the points x = 1.4, 1.41, 1.414,…

A: We make a table of values

Q: 1- Evaluate the limit a-Lim(x/(x-1) – 1/ Inx) b- lim [In(cos(x))]* c- 2-VX²+4 /sin'x x->1+ x >0 x>0…

A:

Q: 4-Find the limits of the following x -1 1. lim Vx +3 - 2 Vx² +2 2. lim X+0 3x –6

A:

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…

A:

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: 1- Evaluate the limit a-Limx²-4/(x²-x-2) X-2 b- lim [In(cos(x))]* c-2-(X²+4) ¹/2²/sin²x x-0 x →0

A: L'Hospital rule:- If a function is of the form p(x)q(x)   then if limx→a p(x)q(x)  is of 00 or ∞∞…

Q: Prove that if lim f(x) exists and lim [f(x) + g(x)] does not exist, then lim g(x) does not exist.

A: Given: limx→cfx exists and limx→cfx+gx does not exist, To show: limx→cgx does not exist.

Q: Show that lim e/n = 1 for any constant c> 1. Use the squeeze theorem (or any other method) to…

A:

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

A:

Q: Let f(x) = |x|/x . Prove that limx→x0 f(x) does not exist by using a sequential argument.

A:

Q: 19 let ɛ > 0 be arbitrary. Determine & as a function of e 2 In formally proving that lim -x + 8 1/4

A:

Q: 5.6. Prove that: (b). lim z0 Z (а). lim does not exist, = 0. - ||

A: The given problem is to prove and evaluate the following complex variable limits.

Q: Use the e – 8 definition of the limit to prove that lim x2 – 2 = 2. %3D - x→2

A:

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

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: Prove that if f is continuous, f(x)≥0 for all x and limx→∞f(x)=0=limx→−∞f(x) then f has a maximum…

A: Consider the given statements and conditions regarding f(x). Function f is continuous.So, fx is…

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: Prove, using the formal definition, that lim→-1 (T+1)² +0o. %3D

A: We have to find the limit limx→-13x+12=+∞.

Q: 19 , let e > 0 be arbitrary. Determine & as a function of e In formally proving that lim + 8 2

A:

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

A:

Q: 9 -x2 if x 1 .What value of c makes g(x) continuous at x = 1? Show all conclusions. 2 Vx-4 Using…

A: For a function to be continuos, the left hand limit should be equal to the right hand limit.…

Q: Suppose there is a number B such that |f(x)/x| < B for all x # 0. Then prove that lim f(x) = 0. 1.…

A: Given that: |f(x)/x|≤B

Q: 1.) Determine the limit: a) lim vx2 + 14x + 49 x-5 b) lim (x² – 4x)³ x-1

A: Solution:To solve 1a and b1.(a)limx→5x2+14x+49(b)limx→1x2-4x3

Q: 4. Use the e - 6 definition of limit to show that lim (2: +3i) = 2 + 5i. 2+1+1

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: 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: Use the Squeeze Theorem to show that lim (x² cos 21xx) = 0. Illustrate by graphing the functions…

A:

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

A:

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

A:

Q: 1. Give an example of a function g for which lim g(x) does not exist, but the function I+2021 value…

A: We have to find the required example first.

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

A:

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: 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: Can the Squeeze Theorem be used to determine lim f(x) when 1(x) < f(x) < u(x) on (1,2) U (2, 3) and…

A: We need to choose correct option.

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: 3. Compute the limit algebraically Vx +3x2 lim 6x 1

A:

Q: 2) Show that the limit exists : 3x3 ,3) lim (x.y.z)-(0,0,0) x2 +y² + z2

A:

Q: prove that lim x-2 f(x)=4 if f(x)= x^2, x does not =2…

A: Given,           To prove limx→2f(x) = 4 and f(x) =x2      if  x≠21        if x =2

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: 3. Compute the limit algebraically Vx + 3x2 lim 6x 1

A: Solution: The objective is to find the limit of the given function

Q: Compute the lim as x goes to 0+ (e^x - 1 - x)/ (xe^x - x)

A: we have to compute the limx→0+ex-1-xxex-x

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.