Compute the limit (x,y)→(0,0) of 8xy / 2x^2+ 4y^2 along the following paths.   (a)  Along the y-axis. (b)  Along the line y=3x. (c)  What can you conclude about the limit?

Q: Verify that L'Hopital's Rule applies and evaluate the limit.

A: The given question belonging to the calculus in which we have to find the limit of the given…

Q: What does it mean for a function ƒ(x, y) to have limit L as (x, y) S (x0 , y0)?

A: To show: "a function ƒ(x, y) to have limit L as x,y→x0,y0"

Q: Express the limit as a definite integral on the given interval. cos(x¡) Дх, [Зл, бл] Xi lim бл dx…

A:

Q: |Find the limits. Enter "DNE" if the limit does not exist. 1- cos(5xy) lim (1,9)→(0,0) xy lim…

A: We have to use proper substitute for the first part.

Q: Express the limit as a definite integral on the given interval. -дх, [1, 6] (x;)2 + 8 lim n- 00 i =…

A: Given limn→∞∑i=1nx*ix*i2+8Δx and 1,6 We have to express the given limit as a definite integral.…

Q: n, ]x| fn (x) = - 2m 0, x > 2n | fn (x) cos x dx lim Calculate its limit. What happens if the limit…

A: Write the integral as limn→∞∫-11fnxcosx dx=limn→∞∫-1-12nfnxcosx dx+∫-12n12nfnxcosx dx+∫12n1fnxcosx…

Q: limit x approaches 0, (xcot2x)

A: Limit x-> 0 ( x/tanx ) =1Calculating the limit of the given expression using this result:

Q: What is the limit of (sinx)/x if x approaches to 0?

A: Given: A function limx→0sin xx and x approaches to 0. Since sin xx is an intermediate form of type…

Q: What is the volume of the solid formed when rotating the region bounded by f (x) = 2/xand the x-axis…

A: Given function is f(x)=2√x

Q: Consider the function fcX) = Vx+8 Part A: us.ng a limit, s how thut f'l1)=- Part B: Find the…

A: Using definition of derivatives (limit form)

Q: Find two paths of approach from which one can conclude that the function has no limit as (z, y)…

A: Given fx,y=x4-yx4+y

Q: What is the limit of f(x, y) %3D as x2 + y? (x,y) approaches (0,0) along the curve y = x?

A: Explanation of the answer is as follows

Q: Express the limit as a definite integral on the given interval. [4(x;")3 – 4x;1Ax, [2, 7] lim i = 1…

A: Solve the following

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: Evaluate the limit using the L'Hopital's Rule. 1 - cosh x lim x→0+ ln(x² + 1)

A:

Q: 32? And are

A: {x2                               if x<-2f(x)={x1/5                           if…

Q: Express the limit as an integral. * lim (5(x)2 - 2(x;)³)Ax over [0, 3] i = 1 dx

A:

Q: Express the limit as a definite integral on the given interval. lim [9(x;")³ – 4x;]Ax, [2, 8] i = 1…

A:

Q: Express the limit as a definite integral on the given interva lim [2(xf)³ – 5x¡]Ax, [2, 5] n- 00 i =…

A:

Q: Express the limit as a definite integral.

A:

Q: Express the limit as a definite integral on the given interval. x lim n- 00 -Ax, [1, 6] (x/)2 + 6 dx

A:

Q: Express the limit as a definite integral on the given interval. lim (2-5(x,)² + 6(x;")³] Ax, [0, 4]…

A:

Q: Estimate the limit numerically to two decimal places or state that the limit does not exist.

A: Given: limx→01-cos3xx2 we know that a3-b3=a-ba2+ab+b2

Q: Estimate the limit numerically or state that the limit doesn't exist: sec-{z) lim z¬1+ vz²-1 (Enter…

A:

Q: Express the limit as a definite integral on the given interval. n lim -Дх, [1, 3] (x*)2 + 8 i = 1 dx

A:

Q: Express the limit as a definite integral on the given interval. lim -Дх, [1, 4] (x;")² + 9 i = 1 dx

A:

Q: Identify the limit that represents the limit definition of the definite integral ∫39 6x3 dx using a…

A: From the given indefinite integral, observe that f (x) = 6x^3, the lower limit and upper limit is 3…

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: limit

A:

Q: Q3 :(A): By using "L' Hopital rules "; Find the limits of : (1) limx-ol: 1-cos x - sin²x

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: lim f(z) = I- - 00 ontor oo If the limit

A: problem is related to limit of a function

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

A: a) The limit can be evaluated as.

Q: Express the limit as a definite integral on the given interval. lim n - 00 -Дх, [1,5] (x*)2 + 6 dx

A:

Q: V1+ sin t dt lim

A:

Q: 4x2 [cos (2)* 1. lim X→+00 Find the limit using L'Hopital's Rule

A:

Q: Estimate the limit numerically or state that the limit doesn't exist: sin(6z) lim (Enter DNE if the…

A: The given limit,limx→0sin(6x)x

Q: Find the limit as x approaches 0  2sin23x/3x2

A:

Q: By considering different paths of approach, show that the function below has no limit as…

A: on solving this we get

Q: Find the limit. Enter DNE if the limit does not exist. x²y lim (x,y) → (0,0) x² + 14y2 II

A:

Q: Prove: the limit as x approaches 0 1-cos(x)/x=0

A:

Q: Vx+14-4 Compute the limit analytically without L'hopital's rule: lim X-2

A: Given limit is : limx→∞x+14-4x-2 To find the value of limit.

Q: The area bounded by the curves y = 1 -x2 and y = - x2 + 2 x + 3 , line x = -1 and x = 1 is

A:

Q: Express the limit as a definite integral on the given interval. * n Χρισ lim Σ -ΔΧ, [1, 8] 0-00…

A:

Q: e-x lim X→∞

A: To Find the limit limx→∞1xe-x By using l'hopital's rule. Let  y = 1xe-x Taking log on both side log…

Q: Find the limit if it exist if it does not exist right DNE limit X to zeros in 2x/4x

A: This question can be solved using the concept of limits.

Q: How would I find the limit of the attached equation using L'hopitals Rule?

A:

Q: Express the limit as a definite integral on the given interval. lim 8x; + (x;)² Ax, [3, 8] dx

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

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->1+ . Since the given function is continuous in…

Q: Express the limit as a definite integral on the given interval. lim [9(x,")³ – 4x†]Ax, [2, 6] n- 00…

A:

Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0). The multivariable limit of this function as it approaches the points (x0, y0) and is denoted as: