3. Evaluate the following limits using polar coordinates: lim(x2 + y?) In(x² + y²) (x,y)¬(0,0)

Q: Evaluate the limits in Exercise 7 and 8. 7. lim sin t dt 8. lim x cos t dt

A: (7) ∫-xx sin t dt =-cos t-xx=-cos(x)--cos-x=-cos(x)--cosx=-cos x+cos x=0 So we get limx→∞ ∫-xx sin…

Q: 1. Determine if the following limits exist: 1² + y? (a) lim (1,y)→(0,0) e² +y² – 1 12 sin(y) (b) lim…

A: In the question we have to find limit.

Q: The value of the limit lim is x-0 sin x а) 0 b) 1 1 c) d) ∞

A: Given options are wrong.  Limit will be e^(1/6).

Q: Determine if the following limits exist: x² + y5 lim (x,y) →(0,0) 1 - cos(x) + y² (a)

A:

Q: Use polar coordinates to find the limit. [Hint: Let x = r cos 0 and y =r sin 0, and note that (x, y)…

A: Here, the objective is to evaluate the multivariable limit.

Q: ez+y? 1 - 1 < MVx2 + y? x2 + y? for all (x, y) (0,0) in some neighborhoud of (0, 0). (c) Using part…

A:

Q: Compute the following limit: 1-cos 2z 1. lim I→0 z sin 2x· Hint: 1 – cost = 2 sin? t.

A:

Q: - Determine if the following limits exist. Justify your answers (b) (Rez)(Imz) lim z→1+i z = x + iy…

A:

Q: Find the limit: lim (e-3t i+ t2 j+ cos 2t k sin?t

A: Limt-0(e-3ti + t2j/Sin2t + Cos2t k).

Q: 1. Determine if the following limits exist: x² + y5 lim (x,y) (0,0) 1 - cos(x) + y4 (a)

A: Sol: Part (a) limx, y→0, 0x2+y51-cosx+y4Find two different paths to approach the point that gives…

Q: 6. Compute the following limits. ry (a) lim (z.) >(0,0) 3r² + y6° 3r2 – 7ry + 2y² I – 2y (b) lim…

A: Ans A):

Q: Prove that lim r(x)= where m = 0, 1, 2, 3. .. II

A:

Q: Find the followings limits:- х — у 1. lim (x,y)¬(3,3) Vx – Jy. 1 2. lim х-0 \xуx + 1 X.

A:

Q: Use the Squeeze Theorem to prove that lim (r.y)→(0,0) x +y² 0.

A: Using Squeeze Theorem to prove that lim(x,y)→(0,0)x4yx4+y2

Q: Find the limit, if it exists, or show that the limit does not exist. 1 - y lim ry cos(r - 2y) (a)…

A: we can find the limit of the given function by choosing different different path. Note : As per our…

Q: Show that lim (z,y)→(0,0) 4 z - Va² + y? V16 – x² – y? – 2²

A: Show thatlimx,y→0,0z-x2+y24-16-x2-y2-z2=0

Q: (a) Find the solution of the limit using polar coordinate system. x² + y2 lim (x,y)-(0,0) x2 + y?…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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: Assuming the limit exists, use polar coordinates to find it x²y? lim (1,9)→(0,0) x² + y²

A:

Q: 1. Determine the following limits, or clearly explain why they do not exist. 3 - t+ sint (a) lim L…

A: Hello. Since your question has multiple sub-parts, we will solve first three sub-parts for you. If…

Q: Calculate the following limits, if they exist. lim (7,y)→(0,0) 1 – cos(x² + y²) 1² + y²

A: Use the polar coordinate substitution

Q: Evaluate the limit ´sin(t) -i + In(t + 5)j + (t + 5)* lim t+0 t Enter your answer in i, j, k form.

A:

Q: Question 2 5x limr→-0 g² + 10x – 1 c→-∞

A:

Q: パーy (x.y)-(0,0) x + y 2.) Prove that lim does not exist.

A:

Q: Find the limit. lim (cos(t), cos(t), t In(t)) t-0+

A: Given: limt→0+cost, cost, t5lnt

Q: e-Y sin(y) cos(y) limy→∞

A:

Q: Use polar coordinates to find the limit. [Hint: Let x = r cos(6) and y = r sin(8), and note that (x,…

A:

Q: Use polar coordinates to find the limit. [Hint: Let x = r cos(8) and y = r sin(8), and note that (x,…

A:

Q: Evaluate using the Squeeze Theorem.

A:

Q: Find the limit. lim (cos(t), cos(t), t* In(t)) t-0+

A:

Q: 2. Evaluate the following limits if they exist: sin(x2 + y?) (a) lim (x,y) (0,0) 2x² + 2y2 xy (b)…

A:

Q: Let y(t) be a solution of ý = y(1 - ;) such that y(0) = 4. Determine lim y(t) without finding y(t)…

A: Given that y(t) is the solution of y.=12y(1-y2)with y(0)=4To find the limitlimt→∞y(t)

Q: Evaluate the limit: r(t+h) – r(t) for r(t) = (t4, sin t, -5) lim h→0 h r'(t) =

A: The derivative r′(t) of the vector-valued function r(t) is defined by…

Q: Determine if the following limits exist: x2 + y? (a) lim (x.y)→(0,0) e# + y² – cos(y) IVly[ sin(y)…

A:

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: Find the limits in Exercises 9–12. sin 2x cos 0 9. lim 10. lim 0→- 30 х r + sin r 2r + 7 – 5 sin r 2…

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: 1. Prove the following limits xy lim (x,y)→(0,0) √x²+y² x³+y³ D) E) lim (x,y)→(0,0) F) x²+y² x² - y²…

A:

Q: Find the followings limits- x - y 1. lim (x,y)¬(3,3) Vx - Jy 1 2. lim x→0 \xVx +1

A: Given: 1. lim(x, y)→(3,3)(x-yx-y) 2.limx→0(1xx+1-1x)

Q: Use the Squeeze Theorem to prove that lim væ[1+ (sin(27/x))²] = 0.

A: "Since you have asked multiple questions in a single request, we will be answering only the 1st…

Q: Examine whether the following limit exists. |y| Væ2 + y? lim (x,y)→(0,0) V x² + y +

A: Given:

Q: Find the following limits (or write DNE for any that doesn't exist): а. lim x? + 3x -1 x-1 x2-1 b.…

A:

Q: Let YR(t) = Reit for 0 <t< 1. Prove that %3D 1 dz = 0. lim R00 JYR z3 + 2z You may use Proposition…

A:

Q: 2. Evaluate the following limits if they exist: sin(r² + y²). (a) lim (1,9)(0,0) 2x2 + 2y² Ty (b)…

A:

Q: 4. Evaluate the limit: lim sin arctan x V2 b. 2 1 d. DNE a. C.

A: Evaluate the given limit.

Q: Зу Let flx,y) = , find the lim(xy)¬(-1,1)( Зу %3D x+y' ) along the curve y=x² .

A:

Q: 1 – cos(x² + y?) x2 + y? - lim (x,y)→(0,0)

A: Given lim(x,y)-&gt;(0,0)1-cosx2+y2x2+y2 find out the limit if it exists

Q: cos t – cos² t Evaluate lim t→0 t

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

Q: Prove the following result: lim csc 0 – cot 0 = 0 0→0

A:

Q: Find the limits. Enter "DNE" if the limit does not exist. 5xy + 2yz lim (z,y,2)→(1,1,4) x² + z² lim…

A:

Q: Use polar coordinates to find the limit. [Hint: Let x = r cos(0) and y = r sin(0), and note that (x,…

A: Given,      lim(x,y)→(0,0) x2y4x2+y2

Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.