If lim r(x) =3 then (2) =3 *+2 A) True B) False

Q: is it true that ∇f(limx)=lim∇f(x) and further that ∇2f(limx)=lim∇2f(x)?

A: We know,  ∇f(xi)=f(xi)-f(xi-h), h=xi-x0i,i=0,1,2...And…

Q: then

A: The given statement is False.   The detailed solution is as follows below:

Q: Show that lim(5x+2)=-18 x-> -4

A:

Q: Evaluate the limit. 1 1 lim x--4X + 4

A:

Q: Suppose lim, =1. Prove that lim -(-1)* = %3D

A: Given: limn→∞ an=1To prove: limn→∞ 1n ∑k=1n -1k a2k=0

Q: Prove the lim(Sn) = -∞ case.

A: We use the notation uN =inf{ sn :n>N},              vN =sup{ sN : n>N }, u = lim uN = lim inf…

Q: Find limx->∞,if for all x>5

A: We apply limx->∞ on all sides.

Q: Find the one-sided limit (if it exists).

A: We first factor the denominator

Q: 2. Evaluate the limit: TC 1000 lim x-7- (x-7)2 lim x→8+ (8 - x)3

A: the given limit is: limx→7-πx-72-limx→8+10008-x3 we have to evaluate the given limit.

Q: Find the one-sided limit

A: To find:limx→-4- x2+2x+4

Q: Prove that lim, = 1, (x > 0).

A:

Q: If (x, y)lim(2, 3) f(x, y) = 4, can you conclude anything about f(2, 3)? Explain.

A: Assume the function z = f(x,y).First observe the value of the limit for x followed by y.

Q: Evaluate the limit x – 4 | lim 2--1 8x2 – 3x + 8

A:

Q: Use continuity to evaluate the limit. 32 + Vx lim x→4 32 + X

A:

Q: lim 21/* is no limit 1

A:

Q: What is lim x-inf (x_n) and y=lim sup (x_n) where x_n = [(-1)^{3n}] + 2?

A: We have to find the lim inf and lim sup of sequence

Q: Show that 2n lim n+2 = 2,

A:

Q: Evaluate this limit. x + 2 lim x--2 x2 +x – 2

A: To find the limit

Q: Compute the indicated limits. 5x -3 lim -00 Væ? + 3 5x 3 - lim -00 Vx? + 3

A:

Q: Use the ε- definition of limit to prove...

A: let the value of the ∂ be ∈3 thus the limit limx→2(3x-2)=4 is proved when the expression…

Q: Prove that lim x 2 = 4. x→ 2

A:

Q: Compute the limits that exist. lim x---0, x2 + 3x/x

A: Given: limx→0x2+3xx

Q: if f(5) = 4, f'(5) = 7, then lim of (f(x²+1) -4)/(x²-4) = ?? when x goes to 2

A: Please see the answer below.

Q: Prove (by using the definition of limit): lim x-> 1(3x - 4)= -1

A:

Q: Find the one-sided limit (if it exists).

A: Given:limx→2-x2x2+4=limx→2- x2limx→2- x2+4=2222+4=48=12

Q: a) Prove the statement lim +3 by using the ɛ, ô definition of the limit x→6 4

A:

Q: Find the one-sided limit (if it exists).

A: Given limit: limx→12-xsecx

Q: How do you solve question 11 and 12?

A: (11)Given:

Q: Find lim *2 x -2 In(x- 1) by using the La'Hospital Rule.

A:

Q: Find the one-sided limit (if it exists).

A: Given :                       f(x) = limx→2- 1x2-43

Q: Use continuity to evaluate the limit. lim xV 13 - x² x-3

A: To evaluate the limit limx→3x13-x2, using continuity. we claim x13-x2 is continuous at x=3. We know…

Q: Prove using the formal definition of a limit that: limx→3 (2x + 2) = 8

A: Given limx→3(2x+2)   = 8Right hand limitlimx→3(2x+2)Let x = 3+h    where h→0hence limh→0(2(3+h)+2) =…

Q: Use continuity to evaluate the limit. 16 + Vx lim 16 + x

A:

Q: Find the limit (if it exists). If it does not exist, explain why.

A: To evaluate the limit: lim△x→0-1x+△x-1x△x. Here, x will be treated as constant. When we put the…

Q: Prove that lim -- (x + 1)

A: we have to prove that: limx→-1-5x+13=-∞   L.H.S:…

Q: Which property of f(x) = x' allows us to conclude that lim x- 8?

A: f(x)=x3 is a continuous function. So, it is continuous at x=2 also.

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

A: Given: 2-x≤f(x)≤2 cos(x)

Q: By using the definition of limit, prove that limx→-3 2x+9 II

A: solution is given below..

Q: find the limit limx -&gt; 3 of [root(x+6) -x] / (x^3 - 3x^2)

A: To determine the limit of given function.

Q: Prove lim flx) - fGc) lim flgth) - f(xa) xうX。 X -X. Hint: x= X,+ h

A: To prove limx→x0fx-fx0x-x0=limh→0fx0+h-fx0h Take  x=x0+hx-x0=h

Q: 3. Show algebraically that lim x* – 2 – x = 0

A:

Q: 5, evaluate the limit. 2x3-x 5) lim X0 x2+3x-1

A:

Q: Use the –δ definition to prove that lim x→1 (3x + 5)/2 = 4.

A:

Q: 2) If 2 – x2 < f (x) < 2+ 3x² whenever -1 < x < 1, use limit theorems to find lim0 f (x). Give all…

A:

Q: Use continuity to evaluate the limit. lim XV13 - x2 X-2

A:

Q: ƒ (₂) If lim fx x-0 = L then which of the following must be true?

A:

Q: Find the limit. 4et – 4 1 + t – 1 2 - - lim 1 + t,

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

Q: Prove lim x ---> 3    (1/3x+1) =2

A: It is known that if we have a function f(x), and limit x tends to a, then the function can be…

Q: Prove the statement using the €, & definition of lim(x² + 2x - 7) = 1. →2

A:

Q: he limits of the followi X-2 2. lim х->2 X — 8 x'-8

A: