16. lim, (1+) =c, then lim, (1 + =? A. 2e B C.eD. e В. X→8∞

Q: Show that (2o + h)" – 2ö = nz- | n-1 lim h→0 h for any zo E C.

A: To show: limh→0z0+hn-z0nh=nz0n-1for any z0∈ℂ

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: 5. If b > 0, prove that lim = 0. %3D 1+nb

A: as per our guidelines we are supposed to answer?️ only one question. Kindly repost other question as…

Q: 2. Evaluate the following: Va2-x² So Sa* x² ydydx а. xsin x b. lim-2cos x+e

A:

Q: (b) Show that if lim sn = +o∞ and lim tm > -x, then lim($n + tn) = +0.

A:

Q: 7 sin(T2) In (zº) q(z) = %3D What is lim q(x)? ncwor bolow:

A:

Q: 4! find lim, B4,-2) S x approachesto I from postve direction.

A:

Q: 3) Using the appropriate approach covered in Unit 6 of this Course along with knowledge ained in…

A: That are application of LHospital rule pf limit.

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

A:

Q: 2.14 Show anx" + ... a12 + ao lim n an bn .. provided that b, # 0.

A:

Q: 8. Calculate the following limits. 3 + 4x² + 2y? xy – 2x – y + 2 (a) lim 2x2 + y? (b) lim…

A: (a) Limit does not exist. (b) The limit is zero.

Q: Show: ( (|fl2 – €)² < lim sup E If (n) < (I|fl2 + e)² , Where ƒ E C(R | Z; , c) n=-N

A:

Q: S* 56sin(t")dt Evaluate the limit lim .8

A:

Q: Compute, from first principles, without using L'Hopital's rule eh – 1 lim = 1 h-0 h

A: (18) We have to evaluate the limh→0eh−1h. We know that the expansion of eh, which is given by…

Q: 1. Prove lim, n+2n³+7 2 0, using the definition.

A: I have used only definition of limit of sequence

Q: Show that if xn ≤ yn ≤ zn for all n ∈ N, and if lim xn = lim zn = l, then lim yn = l as well.

A: Given xn≤yn≤zn And also given lim xn=lim zn=l

Q: Suppose that a, > 0 and converges. lim na, = 0. Prove that Ea, n 00

A:

Q: 1. Show using sequences of functions that lim -nz? e dx 0.

A: Solution is below

Q: to evaluate the limit. 3 1 v 78. lim x→ x Jo im In(1 + e') di | In(1 + e') dt 2

A: Answer is  1/2 Explanation is given below

Q: He sold his stock when t = 4. Compute the following (if it does not exist write DNE): Lim-4- J(t) =|…

A: Given,

Q: Find (if exists). lim (х,у) — (0,0) (4 + (x² + y²) In(x² + y²; Dose not exist O 4 0 1 0

A:

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

A:

Q: y om U to 1.0 how can you find limits bntegroutlon.

A:

Q: 3. Show that if x > 0 for all n e N, then lim(x,) = 0 if and only if lim(1/x,) = +oo.

A: Consider the given information. xn&gt;0 and limxn=0 Now, use the limit definition of the sequence.…

Q: (7) ifa lim has a point that the equalio- prove PCz,y) is above L and only if 3x +2y >4

A:

Q: Show that when T(z) = az+b cz+d where (ad - bc ‡ 0), find lim T(z) = ∞ if c = 0. 7-3

A:

Q: 18. Find (if exists) * lim (x,y)→(0,0) (4 + (x² + y² ) In(x² + y² )) O 1 O Dose not exist - O 4

A: Limit of the two variables function : Let f be a function of two variables .The limit of f(x,y)…

Q: e n Show that for any &> 0 and S, ₁X₁, lim P(S₁-0 ≥ 8) = 0. =

A: We will use Chebychev's inequality to find the required limit of probability.

Q: Txーt e lim 4x² 4,2 +4x

A: Given that limx→0 e2x-14x2+4x We have to find the limit using L-Hopital rule L-Hopital's rule: If…

Q: OF# Usc the L'Hospital's Rule to compute the limits: B) lim x+5 B00 2x+5* A) lim log, (1+ 2x+x?) and…

A:

Q: (3 + 5y +7{ ) 2+y? 200 + y02 exists if lim r)(0.0)

A: This is a problem of multi variable calculus.

Q: How can I find A, k, and C with the following information? L(t)=-(A/k)e^(-kt)+c Lim as t approaches…

A: Given function,

Q: 2. The value of lim is (z.v)-(0,0) Vr² + y? (b) 1 a) 0 (c) 2 (d) does not exist

A:

Q: (ii) 2tan?x lim- x2

A: Given that: limx→02tan2xx2

Q: 5. Use the concept of limits to solve for the following problem iz3 – 1 lim z-1 z + i

A:

Q: Inz lim m>0is Select one: a.() a.0

A:

Q: Examplesa using the definitin of limait to .2 2. Prove i lim Z =Z

A: In mathematics, a limit is the value that a function (or sequence) approaches as the input (or…

Q: (4x – 5y) lim (x,y)→(1,1) 5x² – 7y O -1/2 O 1/2 Limit DNE

A:

Q: 8. lim In(+h)–1 A. 0 B. - C. 1 e D. e E. Does not exist

A: To solve the limit.

Q: %3D In(1+x) 3.A) lim P-ove tha t マー0

A:

Q: Could you use L'Hopital's rule to solve the following question in its current form: lim,0+ In(x+1) O…

A: We have one of the following form limx→af(x)g(x)=00 or  limx→af(x)g(x)=±∞±∞  In these cases we have…

Q: S 56sin(t)dt Evaluate the limit lim

A: Evaluate the limit limx→0 ∫0x56sint6dtx7

Q: (e) lim e* ニD 2. (g) lim -X- www. indetern

A:

Q: 2. Let x E (0, 1), use e – d to prove V4 +x – 2 lim 1 4

A: 2. Given x∈0,1, limx→04+x-2x=14                       ... (1) Use ε-δ definition to prove above the…

Q: 18. Compute, from first principles only, without using L'Hopital's rule eh – 1 lim h-0 = 1

A: given is  limh→0  eh-1h=1we know    eh  = (1+h1!+h22!+h33!+.............)limh→0  eh-1h = limh→0…

Q: Find the limit. lim ey sin P -(5, 0, -5) O 1

A:

Q: (i) Prove that Lim Vent +1 1 1 + + + /2n²+n] ... V2n2+2

A: Given that, g(x)=limn→∞12n2+1+12n2+2+....+12n2+n from sandwich theoram we can say if f(x)≤g(x)≤h(x),…

Q: Enter a whole or decimal number, or DNE for does not exist or undefined. |0x| lim (5+10=¹). [ - -0+z

A: Topic = Limits

Q: 1. Use L'Hopitals's rule to find lim c In(e" – 1). |

A: limx→0+ xlnex-1

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.