Which of the following expressions are true for the limits : کور ua )"د 2(, Зх- 2y e*+y I. II. lim (х,у) >(0,0) 1+ е*-у III. lim (x,y)-0,0) (x² + y²).sin- lim (x,y)-X0,0), x² +y? +y? (a) I: Limit does not exist (b) I: Limit does not exist 1 II : Limit exists and the value is II: Limit exists and the value is 2 III: Limit exists and the value is 1 III: Limit exists and the value is 0 (c) I: Limit exists and the value is - (d) I: Limit does not exist 1 II: Limit exists and the value is 2 1 II: Limit exists and the value is 2 III : Limit exists and the value is 0 III : Limit does not exist (e) I: Limit does not exist II : Limit does not exist III: Limit does not exist

Q: What is lim x=inf (x_n) and y=lim sup (x_n) where x_n = [(-1)^(3n)] + 2? (express answer as x,y)

A: Solve the following

Q: Use the definition of a limit to show that (i) limx→0 (x^2 + 6x + 9) = 9 (ii) limx→−7 and…

A: Authoring guidelines:…

Q: use the definition of limits atinfinity to prove the limit. {lim x approaches to infinity  ( 2/…

A: The function is . The limiting value of the function is to be found out as tends to infinity. To…

Q: Find the limits in Exercises 13–24 by rewritin - 2xy + y² x - y 13. lim (x, y) (1,1) x+y 14. lim (x,…

A: The given expression is limx,y→1,1x2-2xy+y2x-y

Q: Show that the following limit ry lim (.v)-(0,0) 22 +y? does not exist.

A: Substitute y=mx, and check by applying limit.

Q: Find the limit. Enter DNE if the limit does not exist. lim (2,v) → (0,0) æ² + 14y²

A: We’ll answer the 1st question since we answer only one question at a time. Please submit a new…

Q: What is lim x-inf (x_n) and y=lim sup (x_n) where x_n = [(-1)^{3n}] + 2? (express answer as x,y)

A:

Q: 2. Show function f(x,y) = does not have limit of a function at point (0,0). 2² + ?

A: We need to show that the function does not have limit at point (0,0)

Q: Find the limit z3 + 4'iz? lim z→4i z2 + 4?

A: We will simplify given function using factoring it to find limit

Q: Use the Squeeze Theorem to show that lim x? cos (Hìnt: cos < 1 = 0

A: We have to find

Q: How does the sandwich theorem enable us to calculate a variety of limits?

A: Sandwich theorem: Consider 3 function f(x),gx,hx such that fx≤gx≤hx. Also, assume that limx→cf(x)=L…

Q: lim 2tan (2.x) 3n 4

A: This is a basic limit problem

Q: Find the limit. Enter DNE if the limit does not exist. lim (x,y) → (0,0) x² + 12y?

A: Given limit, lim(x,y)→(0,0)x2yx2+12y2

Q: Use the properties of limits to calculate the following limit. Зху - 3 lim 2 (x,y)→(3,3) x + y +5…

A:

Q: 2. Determine if the following limits exist. Justify your answers. (Rez) (Imz) (b) lim z→1+i z = x +…

A:

Q: limit as x----0 for the function (px)/(sqrt(bx+a^2))-a using L'Hopitals

A: Apply the limit.

Q: find the limit of ƒ as (x, y) →(0, 0) or show that the limit does not exist. ƒ(x, y) = y2/(x2 + y2)

A: Let f be a function of two variables, limx,y→0,0fx,y The value of limit exist if, limit for…

Q: Use two different paths to demonstrate that the limit: lim(7,9)→(0,0) x*y?+(x=y)?

A:

Q: Prove that lim (r2- 1) = 3 using the e, 8 definition of the limit. %3D 141 Linh tL

A:

Q: 2ry? does not exist. now that the limit lim (z,y)+(0,0) x² + y!

A:

Q: evaluate the limit using continuity lim(x,y)→(−2,1)2x2/4x + y

A: To evaluate lim(x,y)→(-2,1)2x24x+y

Q: lim Z-y)-(0,0) z4+y?

A: To determine the limits of a multivariate function , Convert the limits into one variable and then…

Q: By considering different paths, show that the limit doesn’t exist. lim (x,y)→(0,0) x2yey/ x4+4y2

A:

Q: Compute the following limit along lines of the form y = mx, for m 0, and along the parabola y= x.…

A:

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

A:

Q: Explain why lim (csc 0 – cot 0) involves an indeterminate form, and - then prove that the limit…

A: limθ→0cscθ-cotθ By applying the limit directly, we get limθ→0cscθ-cotθ=csc0-cot0=∞-∞ Which is an…

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A:

Q: What does it mean for a function ƒ(x, y) to have limit L as (x, y) → (x0 , y0)? What are the basic…

A: What does it mean for a function ƒ(x, y) to have limit L as(x, y) → (x0 , y0)? What are the basic…

Q: Choose the limit that would NOT use l'hopital's rule. 1³ – 1 lim I →1 1² – 1 (=* – 1) (=² – 1) * lim…

A: We can find the answer as given below

Q: Find the limit 10xy lim (2,y) → (21, y1 ) 25x2 y2 at the following points. - 10xy lim (a,y) → (0,11)…

A:

Q: lim tan {x,y}→{0,0} lyl

A:

Q: Show that the limit doesn't exis 3x2 1) (x,y)→(0,0) x² + y² lim 2) (х,у

A:

Q: A function h(x, y) is defined by h(x,y)=(x^2 y)/(〖7x〗^6+y^3 ). Verify the limit over h(x, y) exists…

A: Find the limit approaching the origin along x-axis i.e. y = 0.…

Q: Evaluate. (a) limit as x goes to 0 -&gt; (tanx)/(2x) (b) limit as x goes to 5 -&gt; (-1)/(x-5)2 (c)…

A: a) The limit can be evaluated as.

Q: xãy – xã lim (x.y)-(0,1) x³ + (y – 1)*

A: Solution...

Q: By considering different paths, show that the limit doesn’t exist. lim (x,y)→(0,0) xy4/ x2+y8

A: Answer

Q: Explain why f(x, y) must approach a unique number L as (x, y) approaches (a, b) along all paths in…

A: We have to explain why fx,y must approach a unique L as x,y approachesa,b along all paths in the…

Q: lim(x, y) – (5,9)(÷ ) = ;

A: To show                limx, y→5, 9xy=59 using the epsilon-delta definition of limits. Pre-requisite…

Q: x²y lim (x,y)→(0,0) x2 + y?

A:

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 limit sin 2z cot 8z lim z→0 z cot 4z 16 2 does not exist 4 3/2 O 1 O

A:

Q: Find the limit: 4z4 – 4y4 lim (1,9) → (3,3) 4yª z² - y?

A: lim(x, y)→(3, 3) 4x4-4y4x2-y2

Q: xy? lim (x,y)→(0,0) x² + 2y4 Compute the limit

A: Given limit,

Q: lim (x.y)>(0,0) 2x sin y 2x +y Use the &- d definition of the limit to prove that

A:

Q: Find the limit of f(1,y) = as (r, y) approaches (0,0), if it exists. If it does not exist, show why.…

A:

Q: Estimate the limit numerically or state that the limit doesn't exist: sec (2) lim (Enter DNE if the…

A: To estimate the limit of the given function at x-&gt;1+ . Since the given function is continuous in…

Q: Prove using the def. Of a limit (b) limx→0(2x^2) − 3)) = −3

A:

Q: Find the limit if it exists.   lim(x,y) ---> (0,0)     (x4-16y4)/(x2+4y2)

A: Given: limx,y→0,0x4-16y4x2+4y2

Q: Calculate the limit for the given function and interval. Verify your answer by using geometry.

A:

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.