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

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

A: Limit exist

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

A:

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: 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: 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: Draw the phase portrait for i=2 -1. If z(0) = 1/2, what are the limits lim z(t)?

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: 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: 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: Q2: (a) By using " L' Hopital rules "; Find the limits of ::- (1) lim,-(7/2)-(tan x)" (2) limy-o…

A:

Q: V5+s-V5 lim s→0 S

A: lims→ 05+s-5s=lims→ 01s+5+5Plug in the value s=0=10+5+5=125

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

A:

Q: 5. Conclude on the limit of f(x, y) = +v4 xy2 as (x, y) → (0,0) along paths y = mx and x = y?

A:

Q: Find the limit of the function f(x, y) = 4e10z–3y (x, y) → (0,0). as lim (x,y)→(0,0) 4e10x-3y

A:

Q: (a) Show that the limit xłycos y lim (x.y)(0,0) 2x4 + y2 does not exist.

A:

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

A:

Q: Evaluate this path limit: 4.xy3 =??? lim (x, y)¬(0, 0) 2x² +y6 x=y3

A:

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

A:

Q: 2-メ Compite lim %3D Xラム z(T-X)

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: Let f(x, y) = xy /(x 2 + y2). Show that f(x, y) approaches zero along the x- and y-axes. Then prove…

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: Find the limits in Exercise

A: Given : limh→055h+4+4

Q: Find the limit. x²y lim (x,y) → (0,0) x² + 8y²

A: limx, y→0, 0x2yx2+8y2

Q: (b) Show that lim (x.y)(0,0) 2x6 + y2 does not exist by letting (x, y)→(0, 0) along the curve y =…

A:

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: Find the limits. Enter "DNE" if the limit does not exist. lim cos (x.y)¬(0,0) x' – y' x² + y² 4x lim…

A: Find the limits : Given :            lim(x,y)→(0,0)cosx7-y7x2+y2               =  cos 0-00+0…

Q: X,xif limF,(t)= F(t) at all t for which Fis Continuous

A: Thanks for the question :)And your upvote will be really appreciable ;)

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

A:

Q: By considering different paths, show that the limit doesn’t exist. lim (x,y)→(0,0) xy cos y/ 3x2+y2

A: Answer

Q: Find the limits. Enter "DNE" if the limit does not exist. Va - Vy+7 lim (1,y)→(11,4) Ity+7 х — у — 7

A:

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

A:

Q: 1. a. Describe the level curves of the function f(x,y) = %3D x* + y b. Show that the limit lim (2,y)…

A:

Q: Q2: (a) By using " L' Hopital rules "; Find the limits of ::- (1) lim,-(n/2)-(tan x)cosx (2) lim,--0…

A: L'Hôpital's rule states that for functions fx and gx which are differentiable on an open interval I…

Q: compute dy/dx using the limit definition.

A: Plug the given equation in the limit definition

Q: . Apply the two path test to make sure the following limit exist or not at (0,0) xy f(x, y) x³ +…

A:

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

A: We need to evaluate the following limit limx,y→1,14x-5y5x2-7y

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

A:

Q: Find the limit, if it exists, or type DNE if it does not exist. eV(1z²+1y²) _ lim А. (x,y)→(1,–2)…

A: Given the limits function

Q: Q2: Evaluate: 2 In cosy, [1] limy-o [; y4

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

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: 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: S (2t)*dt lim x→ =? х* — 1

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: By using The definilion of the D Lim x (X+1)= o 2) Lin x Cos

A: Given: 1)limx→0x(x2+1)=02)limx→0x cos1x=0

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

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

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

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),…