6) lim f(x) x-1 f(x) A) 1 B) -1 C) 0 D) Does not exist

Q: 1-z = ? What is lim %3D !- 2!- -2

A: In order to evaluate the limit of a function f(x) at certain point a then replace x by a to get…

Q: Evaluate using the Squeeze Theorem

A: To Evaluate: limx→0xsin1x2 Concept: Squeeze Theorem:limx→afx≤limx→agx≤limx→ahxand if…

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: Formally prove that lim 2x+3=7 using the ɛ,8 definition of a limit. x→2

A:

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: prove limx->a 1/x = 1/a with epsilon-delta definition

A: We need to prove the given problem using the epsilon-delta definition.

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: Prove using formal definition. (a) lim x→0+ 1/x = ∞.

A: Draw the graph of the function .

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

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

Q: Prove that lim x cos - = 7 %3D 1= 0.

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: 8. Using squeeze theorem, show that the following limit (x – 1)² In x = 0. lim (x,9)→(1,0) (x – 1)²…

A:

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: Use two different paths to demonstrate that the limit: lim(7,9)→(0,0) x*y?+(x=y)?

A:

Q: lim -1 x→0+

A: Separate the limit.

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: 5. Use the definition of limit to prove that lim[(x – 1)(sin x²)] = 0

A:

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

A:

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

A:

Q: Show that lim sin( does x→0 x2 .not exist

A:

Q: Prove that lim,→3 3x – 7 = 2 by using e, d definition of limit.

A:

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

A: We have to prove the given limit.

Q: Calculate the limit limx→1sin(x−1)x4−1.

A: value of sin(x) at x=0 is: sin(0) = 0

Q: x² - y? lim ) - (0, 0) х? + у? y

A: Here we have to show that, lim(x,y)→(0,0) x2-y2x2+y2

Q: Q. Prove that: Lim tanhx =1 .

A:

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: 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: 4 Using the e- 8 definition of limit, prove that lim I-3 3r 1 2 -

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: Prove that lim (2x – 7) = 3 by using the e, 8 definition of a limit. x→5

A:

Q: Prove that lim r = 2 using the e-8-definition. Hint: VI - 2 = VT + 2

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

A:

Q: 2) G VALUATE THE UMIT, IF IT EXISTS. Lim X-3 x2-X-12 uT x²+ 3x 2)

A: Given:         limx→-3x2+3xx2-x-12   Factoring                   =limx→-3xx+3x-4x+3

Q: 5. Using the appropriate precise definition of limit, prove that limx-→4+ +0o. Show all work. X-4

A: Here, we need to show the limit by the precise definition of limit. We are given that…

Q: Prove that lim Vr = 2 using the e-d-definition. Hint: VI - 2 = x - 4 %3D VI+2°

A:

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

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: Prove using the def. Of a limit (b) limx→0(2x^2) − 3)) = −3

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(3x–10) =-4 using the ɛ, d definition of a limit. 2

A: Recall: ε -δ Definition :Let f be a function defined in a deleted neighbourhood of 'a' . Then a real…