x+ y Let f(x) = and ɛ = 0.02. Use definition of limits 2+cos x to find 8 > 0, such that for all (x,y) Vx² + y² < ô |f(x, y) – f(0, 0)| < ɛ.

Q: 6 (b). Let 0 < f(x) < cos x and 0 < g(x) < sin x for x E I. Show that there is ro E I such that…

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Q: (a) Let f(x) = tanx- 2, g(x) = x5 + x³ and h(x) = Vcsc5x. Then find %3D the f(g(h(x))), using chain…

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Q: Q3 Limit 4xy78 Consider f(x, y) lim (2,y) (0,0) f (r, y)? What is O No limit Ò 78 O cos 0

A: As per guidelines we have to solve one question

Q: Let f(x) = (1+sin?x) then Limit of f(x) approached to zero equal to : %3D as x Awww

A: Given: fx=1+sin2xx2 For evaluating given limit, we put x=0 in given expression

Q: Discuss limit(n →∞)fn on B and C for  fn(x)=x/n; B=E1; C = [a, b] ⊂ E1 fn(x)=(cos x + nx)/n; B=E1

A: Note: Since you have not mention anything about the set E1, We are considering it any sunset of real…

Q: f(x+h)–f(x) For f(x) = V5x + 20, find f '(x) using the definition f '(x) = lim f '(x) Using this,…

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Q: lim f(x, y) does not exist, where (1,4)→(0,0) Show that f(r,y) (x² + y1)³

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Q: If f isa differentiable function satisfying f(0) = 0 and f (0) = 2, then 1- cos( f (x)) lim x-0 A) 0…

A: The given limit is, limx→01-cos(f(x))x2. Note that, the given limit is of the form 00. Apply L…

Q: If f(x) = xtan(), then f'(x) = %3D (A) (In(x) + tan(e)) xtan(2) (B) sec²(x) In(x) ætan(z) (C)…

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Q: 1 Let f (x, y) =x²y*sin (- x²+y* when (x, y) ± (0,0) and f (0,0) =0. determine if f is continuous in…

A: We have to show given function is continuous at origin

Q: II. Find all values of a and b that make -a -x, x<-2 f(x) = {bx² + 1, Vx + b, -2 Sx<1 x21 continuous…

A: Given:            fx=-a-x,       x&lt;-2bx2+1,     -2≤x&lt;1x+b,       x≥1 Continous everywhere:  at…

Q: Show that limit t  appraoaches infinity  integral -t  to t  x dx =0  This shows that we can’t define…

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Q: 2. 지2 f f Š{(») = 6 , evaluate f(2sin6) Coso d@ is continuous and Sf(x)

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Q: Q4:- Exam the limit of f(x,y) (xy) (0,0) as (xy)(0,0) along the line y = mx, and along the parabola…

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Q: Solve the IBVP for heat equation. Also find the limit of u(x,t) as t -> inf ut = 3uxx + F(x,t),   0…

A: We are given the following system of heat equation. ut=3uxx+F(x, t),   0&lt;x&lt;1,  t&gt;0u(0,…

Q: 1. Let the function f : R → R be given by 1- cos(x) if x + 0, f(x) := if x = 0. 1 Prove that f is…

A: Given : f'(x)=1−cosxx  for x≠0f'(x)=0    for x=0

Q: 10. Find. showing work, the critical numbers of each of the following functions: f(x) = x'et* f(x) =…

A: In this question we have to find the critical points of the given functions which we can find out…

Q: Let f : R → (0, +∞) be differentiable. Suppose that f' has no zero. If lim→0 f(x) = %3D 0, then…

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Q: x2 Consider the function f(x, y) = 2y lim (x,y)(0,0) f (x,y) does not exist by using e – 8…

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Q: u – 12uzx = 0, e00, with the boundary conditions Uz (0, t) = 0, Uz (T , t) = 0, t>0; and the initial…

A: As per the question, we will solve by separation of the variable method.

Q: 15.1. Using the results of the preceding exercise, show that the function g, defined on R to R by…

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Q: 1. Consider the function f defined on [0, 0), f (x) = {x* sin- x + 0 x = 0 where r > 0. Determine…

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Q: Define h(x) = x³ sin(1/x) for x + 0 and h(0) = 0. Prove that h' is not differentiable at x = 0. .

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Q: (c) A non-constant function f : [0, 1] →R such that f(x) > 0 for any r € [0, 1] but | f(x)dx = 0.…

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

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Q: 4. Determine if g is differentiable at x = 0 if 2 cos-1 x, if -1<x < 0 g(x) = %3D if x 2 0. x + cos…

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Q: Please show all work

A: According to the given information it is required to express the limit as a definite integral. The…

Q: be* + a + 1, x 1

A: The given function is  fx=bex+a+1,  x≤0ax2+bx+3, 0&lt;x≤1acosπx+7bx,  x&gt;1 To find the values of a…

Q: QI// Let f(x)= (1+sin?x) x2 then Limit of f(x) as x approached to %3D zero equal to:

A: Apply the limit to the function. limx→0f(x)=limx→0 1+sin2xx2=1+sin2002=10=Undefined Since the value…

Q: _x?yesin (xy2) x2+y2 -Let a function g be defined on R2 at g (x, y) Use: g(x,0), g (x, kx), g (x,…

A: We g(X,y ) first

Q: 1-cosx , x*0 is continuous at x=0 Find k so that (x) =; 4x2 Ik, x=0 A 1/2 в) 1/8 (с) 3/4 D) 2/5

A: Use the definition of continuity to solve a given problem.

Q: 7. Let f(2) = f(x, y) = + i , = 0. when z +0 and let f(0) %3D 5xº+y² (a) Show that lim f(z) = f(0) =…

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Q: (x) = x and f2 (x) = sin (x) sin Wron

A: f 1 (x) = x and f2 (x) = sin (x) sin Wronskian functions that are linearly independent show using.

Q: The limit n lim E V2x; + (x; )²Ax can be expressed as a definite integral on the interval [1, 8] of…

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Q: 19. I. lim g(x) = 5. z-6 II. g(6) = 5. 20. I. g is continuous at x = 6. II. g has a removable…

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Q: 9 The function f is defined by f(x) = 2-3 cosx for 0 <x< 2n. %3D va lues (i) State therange of f

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Q: 2. Consider a function f(x) = cos a² + (5)². %3D (1) Show that f is continuous on R. (2) se the fact…

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Q: (3) Let f : R –→ R be a differentiable function everywhere. Prove that if f (x) = 0 has n solutions,…

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Q: n Express the limit lim > (6(x; )´ – 8(x;)°)Ax; over [2, 4] as an integral. - i=1 . | f(2)dz.…

A: As limit n tend to infinity, del x change to dx.And summation become integration.

Q: a) Evaluate in terms of Gamma function ∫e^(−y^2) y^13 dy   limit 0 to ∞ b) Find ∫f(x) dx limit: 1…

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Q: Find the absolute extrema of the function on the closed interval. f(x) = cos 7EX, = COS 0, minimum…

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Q: Find the following limits proving whether they exist or do not exist 5y¹ cos²x x² + y² C) D) lim…

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Q: dx For = In[F(x)]+C•What is f(/3) if C = 0? (x+x³) (x Tan-lx+ 1) -1 + - - 3. -1 + - + S-is-isS

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Q: Express the limit as a definite integral. [Hint: Consider f(x) = x*.] n lim n5 i = 1 1 7n 2n dx

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Q: II. Given: ‘sinh r − 1 if x 2 x-2 Determine if f is continuous or discontinuous at x = 0 and at x =…

A: Types of discontinuous of the pice wise continuous function

Q: 1-x², x 2 at x = 2.

A: Given, fx=1-12x2, x&lt;23,    x=2x-2-1,   x&gt;2

Q: 3. Let f)=x+x+2x-1E2,lx). Then x= 1 is a root for r() of mmltiplicity 3D1 None 4 2

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Q: a non zero valie for the constant k that makes the fmction ww m tan(kx) x< 0 f(x) = 3x + 2k?, x 20…

A: To find a non zero value for the constant k that makes the function…

Q: Use the properties of continuous functions to explain why f(x) = x cos(x)+1 is a continuous function…

A: y=x and y=cos(x) both are continuous functions  So the product of two continuous functions is a…