Compute the following limit along lines of the form y = mx, for m 0, and along the parabola y= x. What can you conclude? 20ху lim 3 (x,y)-(0,0) X° + yx ..... Along the line y = mx, the limit lim f(x, у) 3D |. Along the parabola y = x, the limit (х, у) — (0, 0) lim f(x, y) =. Therefore lim f(x, у) (X, y) (0, 0) (х, у) — (0, 0)

Q: Evaluate the limit using L 'Hospital's rule: lim x to 0 e^x -1/sin(12x)

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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: Calculate the following limits, if they exist. 2ry lim (z.9)→(0,0) 1² +y²*

A: The limit value of lim(x,y)→(0,0)2xyx2+y2. To find the value of fx,y=2xyx2+y2 when x,y approaches…

Q: Determine if the pointwise limit lim_(n→∞)(ex/n/n) for x ∈ R is uniform. If yes, is the limit…

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Q: Show the limit does not exist. lim (1,y)→(0,0) x6 + y²

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Q: Evaluate the limit with either L'Hôpital's rule or previously learned methods. 5t In(t) lim t- o 8 +…

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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, if it exists, or type DNE if it does not exist. A. lim (lz²13y) (z,y) ( 3, 3)

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Q: Evaluate the limit using L’Hôpital’s Rule

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Q: Use the two-path test to prove that the following limit does not exist. x+ 2y lim (x.y)→(0,0) *- 2y…

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Q: 4. Does f(x) = : (0, 00) → R have a right limit at x = 0? If yes, compute the limit. If not, explain…

A: We need to calculate when x->o+ from right side then f(x) exist or not.

Q: Evaluate the limit using a definite integral 2 lim n n 1+

A: Let x = i/n Hence, dx = 1/n when i = 1; x = 1/n; As n tends to infinity, 1/n tends to zero. Hence,…

Q: Approximate the given limit numerically and verify your answer graphically. Round your answer to…

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Q: Evaluate the limit using L'Hospital's rule ex – x – 1 lim X+0 9x2

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Q: Estimate the limit numerically or state that the limit doesn't exist lim h→0 cos(9/h)

A: We will pick some values for x. Then we will find values of y for those xs.

Q: Evaluate the limit using L'Hôpital's rule. limx→0  ex+2x−1/9x

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Q: Evaluate the limit using L'Hospital's rule: x13* lim x→0 13* – 1 -

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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: Find the limit. lim tanh(x)

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Q: evaluate the limit  limit as x approaches infinity of x+cos(x)/ x+1

A: To find,

Q: Find the value of the following limits: 5y 1-limy→o tan 10y 2-limx-0 6x cot3x 7x²+8x 3-limx-0 4x²…

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Q: Express the limit as a definite integral. lim xk Ax [-3, 2] 4-0 k 1

A: Given

Q: e-Y sin(y) cos(y) limy→∞

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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: Verify that L’Hôpital’s Rule applies and evaluate the limit.

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Q: 2. Compute xe" da using a limit.

A: The given which we have to find the integration from 0 to ♾️  f(x) = xex

Q: (4) Find the limit by interpretting the limit as as an appropriate derivative. (DO NOT USE…

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Q: 9. lim 15 1 + (i/n)6

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Q: Find the limit, if it exists, or type N if it does not exist. 3x2 lim (z,y)→(0,0) 5x² + ly?

A: Solution: The objective is to find the limit

Q: Calculate the limit algebraically. lim (x + Jx)

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Q: Calculate the limit and show proof. lim (1,1)(0,0) r2 + y?

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Q: lim h→0 (e) cos( 4 + h) - 1 h

A: To find the limit of following function

Q: Compute the following limit, or verify that it does not exist: xY lim (x,y)→(0,0) 2x² + 3y2

A: Given that limx,y→0,0xy2x2+3y2 fx,y=xy2x2+3y2 (i) Put x=0 and then y=0 in fx,y find the value of f1…

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

A: Answer

Q: By considering different paths of approach, show that the limit lim (1,y)→(0,0) rª + y² does not…

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Q: Evaluate the limit using L'Hospital's rule. limx→∞(x^5)(e^(−x^2))

A: Given limit is It is of indeterminate from (infinity/infinity),  Therefore, by applying L'Hospital…

Q: Find the value of the.limit Olim Jx+z - V3 X+2 @lim Vx+6 x3-2 X+2

A: A number M is said to be the limit of a function f(x) at x=a if for any arbitrary chosen positive…

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: Use two different methods to find the limit where m > 0, n > 0, and x > 0

A: The given limit is: limx→∞lnxmlnxn where, m > 0, n > 0, and x > 0 First method: To evaluate…

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

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Q: Evaluate the limit using L'Hospital's rule.    limit of 4cos(3x)sec(-7x)  as x approaches pi/2

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Q: Estimate the limit numerically. x² lim x+4 O diverges to +o 00 O diverges to -∞

A: Given

Q: Find the limit by rewriting the fraction first. x- y + 5/x - 5Vy lim Vx - Vy (ху) -- (0,0) x#y x- y…

A: limx,y→0,0x-y+5x-5yx-y =limx,y→0,0x-yx+y+5x-5yx-y =limx,y→0,0x-yx+y+5x-yx-y

Q: Evaluate the limit using L'Hôpital's rule e + x – 1 lim 9x

A: Find the limit

Q: verify that L’Hôpital’s Rule applies and evaluate the limit.

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Q: 8. Let f(x, y) be the function x3 + y3 xy? Show that the limit lim(9)→(0,0) f (x, y) does not exist…

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Q: Find the limit lim, →+∞ In(x² – 1)– In(x+ 1)| Enter I for x∞, -l for -o, and DNE if the limit does…

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Q: 7. Show that the limit does not exist by approaching (0,0) lim .2 ,2 + y- (x, y) (0,0)

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Q: Verify using the definition of limit (with e and 8) that lim 1|+3) = 3, x1 xplain all the steps and…

A: To verify the below limit using the epsilon-delta definition. limx→112x−1+3=3

Q: Find the following limit and show that either the limit exists or not lim (x,y)¬(0,0) xy sin 2 + y²)…

A: claim - to show either the limit exist or not  limx,y→0,0xysin1x2+y2 as we know, limit exist if…