Show that if (sn,tn) converges to (s,t) in ℝ2 with the usual d2 metric, then the sequence converges to the same limit in the d0 ("max") metric.

Q: Use the limit definition to compute f '(a) and find an equation of the tangent line.

A: Answer: f'(-2)=-1

Q: Find the limit (if it exists).

A: If the vector valued function is of the formr(t) = f(t)i + g(t)j + h(t)kThen its limit is given…

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

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Q: Use l’Hôpital’s rule to find the limits

A: Given Data The limit is limy→0 √(ay+a²)-a/y.   Evaluate the given limits by using l’Hospital rule,…

Q: By computing the limit of right-endpoint approximations, prove that b4 x' dx 4 if b > 0, then

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

Q: Find the limit. Enter DNE if the limit does not exist. lim = DNE (7,v) → (0,0) z² + 11y²

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Q: Find the limits. Enter "DNE" if the limit does not exist. x? – y lim (1,y)→(2,2)

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Q: Evaluate the limit In(t + 6) lim e"i + In(t + 6)j+ t – 7 Enter your answer in i, j, k form.

A: Use linear property of limit and use limit separately

Q: 2) Use the definition of the definite integral to write 2 e* dx as a limit. Do not evaluate this…

A: In this we use the limit  definition of definite integral

Q: Find the limit of s(n) as n-* s(n) = Z1+ j =1\ O 17/12 O 14/3 O 10/3 O 20/3 5/6 A Moving to another…

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Q: Evaluate the following limit. sin xy + cos xy 2y lim (x,y) →(0,)

A: The objective is to evaluate the limit.

Q: Find the limit for the following sin(3y).cot (5y) lim y→0 y.cot (4y)

A: we have to find the limit of the following.

Q: Let X = R' and d be the metric on Rº given by d(x,y) = E., \x - y,| where x = (x, X2, Xa) and y =…

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Q: Express the limit as a definite in 7 lim 1 + (i/n)3 n→ 0 n i = 1 1 7n dx 8

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Q: Use Picard's method to find y1, y2, Y3. Determine the limit of the sequence {yn(x)} oo for

A: Introduction: A differential equation is an equation that contains the derivative of an unknown…

Q: Let f (x, y) = 2x 2 + 3y 2/xy. Set y = mx and show that the resulting limit depends on m, and…

A: Given: f(x,y)=2x2+3y2xy

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

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Q: Evaluate the limit. (). lim ( sin i + cos 3' j + tan(t)k

A: Limit (f1 i + f2 j + f3 k) = Lim f1 i + Lim f2 j + Lim f3 k

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: Find the limit. 3 - ti + In(t)j k t-3

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Q: Find the limit: limt t→0⟨(e^5t)−1/t,t^11/(t^12−t^11),−7/8+t⟩ ⟨  ,  ,  ⟩

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Q: Find the limit. lim (cos(t), cos(t), t In(t)) t-0+

A: Given: limt→0+cost, cost, t5lnt

Q: If we approach c from only one side, then we have a(n)____limit.

A: If we approach c from only one side, then we have a(n)____ limit.

Q: Apply the two path test to make sure the following limit exist or not at (0,0) ry f(x, y) x³ + y3/2

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Q: Find the limit. lim (cos(t), cos(t), t* In(t)) t-0+

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Q: use the formal definition of limit to prove that thefunction is continuous at c. h(x) = √,2x - 3, c…

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Q: Evaluate the limit: r(t+h) – r(t) for r(t) = (t4, sin t, -5) lim h→0 h r'(t) =

A: The derivative r′(t) of the vector-valued function r(t) is defined by…

Q: What is the limit of g(t) = t as t → π?

A: We find the limit by substitution

Q: Find the limit, if it exists, or type 'DNE' if it does not exist. eVla²+2y? lim (x,y) (-1,-4)

A: find limit

Q: What is the limit of (In(e+h)-1)/ h at h=0?

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Q: Find the limit (if it exists) of Anx1 as n approaches infinity, where

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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

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Q: Let f(x, y) = 2*. Find fy(1, 2) using the limit definition.

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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: 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 +…

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Q: Show that h, G) = (x+ n) converges %3D x4り? pointwise fo fa)=1, VXE [I, ) ノ

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Q: Compute the limit 1 e¯nt lim dt. VE n-00

A: Given: limn→∞∫01e-nttdt

Q: Use the limit definition to compute f '(a) and find an equation of the tangent line. f (x) = x−1, a…

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Q: Estimate the given limit numerically, if it exists lim, 1* (E) O does not exist O 4

A: Explanation of the answer is as follows

Q: Use L'Hospital's Rule to find the limit as z approaches i of (1 + z^6) / (1 + z^18)

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Q: Express the limit as a definite integral. lim 3 Xk, where P is a partition of [2, 3] k=1.k

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Q: Find the area of the figure using the limit process for: y = x², y=x , x=0, x= 2.

A: Given, y=x2, y=x, x=0, x=2

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

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Q: Evaluate | e* dx as limit of sums. a

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Q: Find the limit of the following function. а sin (aтz) dz

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Q: compute dy/dx using the limit definition. y = 4 − x2

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Q: lim n? +n + 2

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

A: In this question , we can find the limit along the given path, Firstly we can find limit along the…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.