18.V. Let f be as in Definition 18.1 and suppose that the deleted limit at c exists and that for some element A in R° and r > 0 the inequality |f(x) – A| < r holds on some neighborhood of c. Prove that lim f – A| < r. Does the same conclusion hold for the non-deleted limit?

Q: 4] Compute the following limits, using the Algebraic Limit Theorem,where (x„) → 0. 1+2x, a) lim…

A: a) lim 1+2xn1+2xn-3x22 Since limit of xn is zero . Using lim cxn=climxn So, lim 2xn=2limxn=20=0…

Q: 5 Suppose that f: A -R is a function defined on an open interval A, and a E A. (a) Show that if r…

A:

Q: 1. Use the e - ô definition of the limit to show the following: (a) lim(5x – 2) = 3 (b) lim VT = 0

A:

Q: 2. Use the ɛ-d definition of the limit of a function to prove that lim = 3. I-»3 T – 2

A: Let  I  be an open interval containing  c , and let  f  be a function defined on  I , except…

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: (4) Let f(x) be the function whose domain is all of R given by the rule 42 if r € Z and f(x) = 10…

A:

Q: Use the definition of limits at infinity to prove the limit

A: Definition of limits helps to find the limit of the function. A value is said to be the limit of…

Q: Recall that to show lim;-→a f(2) does not exist, we need to find two distinct paths Y1 and y2 of…

A:

Q: 2 Use the definition of continuity to show f (x) = is x+3 continuous at x=1. Whenever you find a…

A:

Q: r-1| limr→1 2r-2

A: We have to find

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)…

A:

Q: Use the Two-Path test to show that the following limit does not exist. (Use y=x2) lim (x,y)→(0,0)…

A: Using Two-Path test we will evaluate.

Q: 3. Prove the following using the epsilon-delta definition of a limit. 1 c. lim = 0 X→+o x-

A:

Q: Find the limit L. Then using the Epsilon Delta definition to prove that the limit is L  lim x→−4…

A: Given,

Q: Prove, using the e-8 definition of limit, that: (a) (b) lim ²+1=2 24-1

A:

Q: 1 Evaluate the given limit of the following: lim [(x² - In(x + 1)) cotx] x-0

A:

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

A:

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

A: It is an application of sandwich principle of limit

Q: 2 Show that the following limit does not exist. lim Ty Cos y (r.9)-(0.0) 3r+ y²

A:

Q: Evaluate the following limit in two different ways: with and without l’Hôpital’s Rule.

A: Consider the provided expression, limx→∞4x4-x2x4+x-1 Evaluate the following limit in two different…

Q: 4. Evaluate the following limit or show that it does not exist. 1 lim 2 u +u,

A: To evaluate the limit,  limu→01u-1u2+u

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…

A:

Q: Use the e - 8 definition of the limit to establish a ô that proves lim (x – 1) = 2 x-2 O A) 3 B) 2e…

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

A:

Q: Compute the following limits, using the Algebraic Limit Theorem,where (xn) → 0. 1+2n a) lim…

A: Consider the given information. (a) Consider the given expression. limxn→01+2xn1+2xn−3x2n Apply the…

Q: 1. Use the definition of the limit to prove the following limit. 1. lim x = 3 x→3 2. lim x +x–11=9…

A:

Q: 2. Give counterexamples to show that these statements are false: (a) If f(c) is indeterminate, then…

A: 2a) This function is indeterminate at x=0  So, we have shown that although f(0) is indeterminate,…

Q: Prove the following limit using the 8,€ definition of lim,→p f(x) = q: lim(z,v)→(0,0) (x² – 2y² + 1)…

A: To prove:: limx, y→0,0x2-2y2+1=1 Proof; For ε&gt;0, consider δ=ε2; As limit is x, y→0,0, then…

Q: 1. Prove, using the e-6 definition of limit, that: (a) (b) lim lim_2²+1=2 24-1 x³+x²+x+1= 4.

A: Limit

Q: Evaluate the following limit in two different ways: with and without l'Hôpital's Rule. 3x* + Vx lim…

A:

Q: 1. Prove the following limits x 4y4 G) lim (x,y) (0,0) (x²+y4)3 = Does not exist

A: Solution is given below:

Q: 1 6. lim x++00 x(Vx* + 1 – x²)

A:

Q: 1 Use the definition of the limit (ɛ – 8) to show that 1 lim = i. z-i z

A: We will prove the given limit.

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: Use the precise definition of a limit to prove the following limits. Specify a relationship between…

A: To Prove - limx→3 x2-7x+12x-3 = -1Cosidering LHS - limx→3 x2-7x+12x-3 = limx→3 x2-3xx-4x+12x-3limx→3…

Q: Use the precise definition of a limit to prove the following limits. Specify a relationship between…

A: Given limx→3-2x+8

Q: 1. By referring to Figure 1; x = 1 x = 3 y = h(x) (2, 0) (0, –) - Figure 1 a. List the x values that…

A:

Q: Prove the following limit. ) f(x)=x²+2x+6 where lim, 3 f (x)= 21 >) lim x – x+1=-5 =) lim x³ = 8

A:

Q: 1. Prove the following using the € - 8 definition of limit. (c) lim ² = 2i. z→1+i

A:

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

A:

Q: 12 Show that the following limits do not exist. 1 lim X0x (a) (x > 0), 4,

A:

Q: Use the definition of the limit (e – 8) to show that 1 lim = i. z-i z

A: Limit Definition: Let f(z) be a function defined in some neighbourhood of z0. A complex number w0is…

Q: 1. Evaluate the following limits by applying the limit theorems (show solutions): 2. 7. x→4 V4x 1 1…

A:

Q: 4] [2M] Determine whether the limit exist in R for lim xsin(1/x²), (x# 0). x→0

A: Zero

Q: 1. Find the following limit, if it exists. lim (z.1)-(0.0) V2 + y? +1-1

A:

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: 1. Consider the function given by if x € QU [0, 1] if x € QºU [0, 1] f(x) = 1- I Determine if the…

A:

Q: 4х + 1 Let f(x) = X-2 (a) Use the precise definition of the limit (i.e. e, 8) to prove that lim f(x)…

A: To solve this question we need to apply the conditions of limit and continuity

Q: 5x2 + 10х + 1 :0. lim x4 + x? + 1

A:

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.