Prove that lim [f(r, y) + g(x, y)] = L, + L2 (K, y) (a, b) where f(x, y) approaches L, and g(x, y) approaches L, as (х, у) (а, b).

Q: find the linearization L(x) of ƒ(x) at x = a. ƒ(x) = tan x, a = π

A: The linearization of a function f(x) at x = a is, Now differentiating  given function with respect…

Q: 2(x-1)* - y² Lex-1)* +2y (d) Show lim does not exist.

A: Limit exist

Q: Find the limit lim Vz +1 Enter I for o0, -I for-oo, and DNE if the limit does not exist. Answer

A:

Q: If f(r, y, z) does lim f(r, y, z) exist? %3D 1² + 2y? + z² ' (z,y,z)¬(0,0,0)

A: To find its limit we have to first convert the function into polar form.

Q: If f(x, y) approaches 1 as (x, y) approaches (0, 0) along the path y = mx, where m is any real…

A: To check whether the given statement is true or false.

Q: Let f (r, y) = 5z'y 7+2y0 %3D Evaluate lim)(0,0) f (x, y). Either find the limit or show that the…

A:

Q: Find the limits in Exercises 1- 3x2 – y² + 5 (x, y)→(0,0) x² + y² + 2 2. (x, y)→(0,4) Vy 1. lim lim

A:

Q: Evaluate the limit or determine that it does not exist.

A: To evaluate the limit: limx,y→0xyx2+y2. Solution: We know that in polar coordinates, x=rcosθ and…

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: x'ey Let f(x, y) : Then lim (xy)-(0,0) x2+3y2 f(x,y) (A) exists and equals 0 since f (x,y)…

A:

Q: Explain why the paths y = mx and x = y² are suitable to check if the limit of %3D xy f(x,y) = as (x,…

A:

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: Investigate whether the limit exists lim (r,y)→(0,0) x²y² + (x – y)²

A: Given :                           lim(x,y)→(0,0) x2y2x2y2+(x-y)2  To find :…

Q: Show that lim (2,v)>(0,0) **+y D0.

A:

Q: Let f (x, y) = xy/(x2 + y2). Show that f (x, y) approaches zeroalong the x- and y-axes. Then prove…

A: Given :                     lim(x,y)→(0,0) xyx2+y2          We know that if the limit exist, all…

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: Consider the line f(x) = mx + b, where m ≠ 0. Use the ε- definition of limit to prove that limx c…

A: We have given function fx=mx+b, where m≠0. We have to prove that limx→cfx=mc+b using ∈,δ definition.…

Q: 2. For f(z, y) = z-y, prove that f is differentiable at the point (1, 1).

A: Given: f(x, y) = x2 - y2

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

A:

Q: 3. Show that lim(r.y)→(0,0) xy =0. (Hint: Use the fact the y| = Vy² and therefore y|< Vx² + y²)

A:

Q: Utt Urr 100, x > 0, t > 0. u (x, 0) = u (x, 0) = u (0, t) = 0 lim u (x, t) = 0, x → 00

A: 1. Laplace transform is an integral transform that converts a function of real variable t to a…

Q: Suppose that f : R -R is differentiable at z = 0. (a) Prove that (t) lim,-o EA = f'(0) (b) Give an…

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:

Q: xY Let f(x, y) : and let L = lim(z,y)→(0,0) f(x, y). Which of the following 3x2 + 8y2 statements is…

A:

Q: Use the Squeeze Theorem to prove lim = 0. x00 x+ e- For x 0, since 0 < e* <1, we have

A:

Q: 1 -1 lím (x,y)→(0,0) tan x² + y?

A:

Q: Let f(z, y) and let L = lim(,1) → (0,0) f(x, y). 5z2 +5y2 Which of the following statements is…

A: We have to check

Q: Consider the function f (x, y) : 2y Show that lim (x,y)→(0,0) f (x, y) does not exist by ɛ – 8…

A:

Q: Find the limits in Exercise

A: Given : limh→055h+4+4

Q: [4] (a) Show that lim (xa)-(0,0) 2x+y? y-mx, or along any parabola y= exists and it's approaches to…

A:

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

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

Q: Find the limits in Exercises 19–22. dx tan-t dt 19. lim b→1", 20. lim VI - x² 21. lim 2n n + 2 n-00…

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

Q: x-y (x,y) →(0.0) x + y 2.) Prove that lim does not exist.

A:

Q: At what points (x, y, z) in space are the functions in continuous? ƒ(x, y, z) = x2 + y2 - 2z2

A: Solution:-  At what points (x, y, z) in space are the functions in continuous? Let (m,n,p) is the…

Q: Let f(2) = C, be the upper half eircle J2| = R in the counterclockwise direction with R>1, and /= ,…

A:

Q: ry lim (z.v)¬(0,0) + y² exists.

A: Please refer the attached image for complete solution.

Q: ove that the limit as (x,y) →(0,0) of f(x,y)=(x²-y2)/(x²+y²) does not

A: Given f(x,y)=x2-y2x2+y2

Q: Find the curl and also the divergence of F(x, y, z) = (x²z, e" + zz, xyz)

A: The objective is to find the curl and divergence of Fx,y,z=⟨x2z,ey+xz,xyz⟩ Curl and divergence of…

Q: Find the limit. lim (х, у) > (1, 55) Vx + у Discuss the continuity of the function. continuous for x…

A:

Q: Shone that lim g (x,yJ-p (0,0) does not exist,

A: According to the definition of limit, we have

Q: Find lim r(t) when 5 In t r(t) = (: 2 tan¬lt, e-2 t 2 tan-t, e 1. limit = (2т, —2, 0)

A: Given    r(t) = 2tan-1t , e-2t , 5 lnttwe need to  find  the limt→∞ r(t)=?

Q: 70. J, f(x) dx may not equal lim , f(x) dx Show that 2x dx o x² + 1 diverges and hence that 2x dx x²…

A:

Q: Find the limit or prove that it does not exist lim (2,y)→(0,0) T° cos x + 2y sin y x2 + 4y2

A: We will evaluate the limit limx, y→0, 0x3cos x+2y3sin yx2+4y2  using polar coordinates.

Q: Evaluate the limit: r(t + h) – r(t) lim h→0 for r(t) = (1-6, sin t, 8) h r'(1) = ( ...

A: We have to find components of r't

Q: Determine lim In/xy. (x,y)(e4)

A: lim(x,y)-&gt;(e2,4)lnxy

Q: Let f(x, y) = 2.ry e" then Lim)-(0,0) (0,0) f(T, y) = %3D A) 4 B) 1 C) 2 D) 0 E) Does not exist

A:

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

A:

Q: F(x) Let F(r) = | 3e" dt. Find lim if it exists. If the limit does not exist, explain why.

A:

Q: Study the continuity of f(x,y) = xe-Vx at p(1,0) ey.

A: