1-x c) lim- x1 1-Vx こ 11- sin x d) lim エー→0

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

A: Solve the following

Q: Use Squeeze Theorem to evaluate lim x→∞ (cos x/ x)

A: It is known that the absolute value of the cosine function is not greater than 1, hence -1x≤cosxx≤1x

Q: evaluate using Theorem 2 as necessary.

A:

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

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

Q: By considering different paths of approach, show that the functions in Exercises 41 have no limit as…

A:

Q: tan 3h 3. Estimate the value of lim,0- by creating a table of function values for h h = 0.01,0.001,…

A: We need to substitute these values of h in function f(h) to evaluate limit

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

A:

Q: 2. Explain why f(x, y) must approach a unique number L as (x, y) approaches (a, b) along all paths…

A: A number L is said to be the limit of a function f(x,y) as (x,y)→a,b , if for any real number ∈>0…

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

A:

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: Use a tble of functional values to evaluate: lim 12 z-2

A:

Q: prove that lim (2X+iy2)=D4i %3D

A: We have to Show given expression of Complex limit.

Q: sec x lim sec(3x)

A:

Q: Show that limx,y)-(-2,0) g(x, y) does not exist given that (x + 2)³y² (x + 2)6 + 3y4' 0, if (x, y) +…

A:

Q: Compute sin ax lim z0 sin bæ

A: L’Hospital’s Rule : Suppose limx→af(x)g(x) = 00  or  limx→af(x)g(x) = ±∞±∞ where a can be any real…

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

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

Q: 2ェ+ COS lim エ→ー 3.x - Vz?+3

A: I=limx→-∞2x+cosx3x-x2+3dividing x b N and…

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>0 there exist…

Q: lim (In w-

A:

Q: Prove limx→∞(1/x)=0

A:

Q: . Use the definition of limit to prove lim = 2r - 3 = 7.. %3D %3D

A:

Q: Is there a value of a for which limx1 3x^2+ax+a+3/x^2+x-2 exists? Enter the value of a if such a…

A:

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

A: Given: limx→0x2+3xx

Q: & Prove that lim I = 2 using the e-d-definition. Hint: Vr %3D

A: Given: To explain the given statement as follows, Here, prove that limx→4 x = 2.

Q: Prove that lim x → 0+ (√x [1 + sin2( 2π/x )]) = 0 (using the squeeze theorem)

A:

Q: Prove that lim x->0+ √ x [1 + sin2 (2 π/x)] =0

A: Given: limx→0+x1+sin2 2πx By using the property of limit, the limit of a product is the product of a…

Q: Use the Squeeze Theorem to show that lim (x2 sin(2)) = 0 x-0 %3D

A:

Q: Verify using the definition of limit (with e and 8) that lim (13r – 1| - 1) = -1, explaining all…

A: Formal definition of Limit. Let f be a function defined on an open interval containing c (except…

Q: Find the limits in Exercise

A: Given: limθ→π4tanθ-1θ-π4

Q: Exercise 3: Use the squeeze theorem to prove lim0 sin(x-1) = 0

A: To prove:

Q: swer of the following lim 2x²+6y² lim (x,y)→(0,0) x²+y²

A:

Q: evaluate using the Squeeze Theorem.

A: We know that sin is always  between -1 to 1 Multiply all sides by (x-1) and then take limit x->…

Q: Using the ɛ, & definition, prove that lim V = 1. %3D x 1

A: We have to prove the given limit.

Q: 1. Prove that lim vx sin O using the Squeeze Theorem.

A: Using squeeze theorem,

Q: Use Squeeze Theorem to evaluate -2x lim e COS x x 00

A: Based on the squeeze theorem or it is also been called sandwich theorem we can evaluate the limit by…

Q: Show that lim |x|cos 0.

A: We have to prove that limx→0xcos1x4=0. Use property, if limx→a−fx=limx→a+fx=L then limx→afx=L. First…

Q: Explain why f(x, y) must approach a unique number L as (x, y) approaches (a, b) along all paths in…

A: We have to explain why fx,y must approach a unique L as x,y approachesa,b along all paths in the…

Q: By considering different paths of approach, show that the functions in Exercises 41-42 have no limit…

A: Hi! You have uploaded multiple questions. As per norms, we will be solving only the first question…

Q: Prove that lim vx sin (2) = 0 using the Squeeze Theorem. x-0+

A:

Q: 6.) Use Theorem 4 lim x csc 3x e 7.) Use Theorem 4 lim e -0 sin? e

A:

Q: Show that lim x|cos )= = 0.

A:

Q: is lim (sec x)ot æ 2

A: Apply exponent rule

Q: Evaluate using Theorem 2 as necessary.

A:

Q: Use squeeze theorem to shoe that limx-> 2 (x-2)2 (3+sin(x2))= 0

A:

Q: evaluate using Theorem 2 as necessary.

A: We write tanx using sinx and cosx

Q: use the squeeze/sandwich theorem to show that lim x-->0 (x^4cos(1/x)) = 0

A:

Q: Prove that lim x-y does not exist. (x.y)(0.0) x + y

A: lim(x,y)-(0,0) (x-y)/(x+y) does not exist.

Q: Evaluate using Theorem 2 as necessary.

A: Given: limx→0 x2sin2(x)

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…