13. Let g be defined by V7x +2 V6x + 4 | g(x) = if x 2 - and x 2 x – 2 k , if x = 2 For what value of k is g(x) a continuous function? Is g(x) continuous at x = -1? Why or why not? NIN

Q: 1. Show that the function, defined for x E R by f(x) = 1 is uniformly continuous 1+x² on R.

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Q: For the continuous function f(x), if f’(-4) does not exist, then x=-4 is a critical point.

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Q: 6. Consider the function f Cx)=5(o+5)-25 if X#し if xfO D.IS fCx) continuous at X-0? EXPlain

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Q: 4 The function f (x) = x 2 2x Select one: a. continuous at x = 2 and differentiable at r = 2. O b.…

A: A function is continuous if Left hand limit is equal to right hand limit.

Q: Suppose f' (x) = g (x) (x + 1) (x – 2)´, where g (s) is a continuous function which is always…

A: When f'(x)>0 then function increasing when f'(x)<0 then function decreasing

Q: 1- Discuss the continuity of the following functions at given points { V9r4+x² 5x2+3x+1 ,if a 0 (a)…

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: Suppose that a continuous function g satisfies: g(x)dx = 42 and g(2x)dx = 3. %3D 2 Find æg(3x²)dx.

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Q: Suppose that f is a function on the interval [1, 3] such that –1 < f(x) < 1 for all x and f(x) dx…

A: It is given that ∫13fxdx=0 and -1≤fx≤1.

Q: Is there a continuous function such that - f: [1,2] U [3,4] → R - f(1) * f(4) < 0 - f(x) not egual 0…

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Q: Suppose that f(1) = 1, f(4) = 6, f (1) = 7, f '(4) = 5, and f " is continuous. Find the value of xf…

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Q: (-x + 1for x 1 h(x) h is continuous at x=1

A: A function is continuous at a point x=a if it satisfies all the three conditions. 1) Function must…

Q: Suppose that f(1) = 4, f(4) = 5, f '(1) = 6, f '(4) = 3, and f " is continuous. Find the value of xf…

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Q: Suppose that a continuous function g satisfies: g(x)dr = 30 and g(2x)dx = 3. 0. 2 Find r9(3x²)dx.

A: Suppose that a continuous function g satisfies: ∫03gxdx=30 and ∫06g2xdx=3. The, we have to find the…

Q: 1. Suppose that f : [0, 0) → R is continuous and f(x) # 0 for all x > 0. If we have (f(x))? = 2…

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Q: Зx2 — 1 if x 1 Determine c and d so that f is continuous everywhere

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Q: Suppose that f(1) = 2, f(4) = 8, f'(1) = 10, f'(4) = -3, and f" is continuous. Find the 4 value of…

A: We have to find the value of the given integral.

Q: 19.D. Show that the function defined by g(x) g² sin (1/x), x * 0, %3D = 0, 0, is differentiable for…

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Q: 3. Find the value of k that will make f continuous on (-o, 00). NORS x + 19 if x 1

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Q: 9. Consider the function f: R → R, defined by (x²sin(1/x) ifx # 0 f(x) = if x = 0 Then a. f is not…

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Q: Find a value of k making h(x) continuous on [0, 5]. kx 0 < x < 2 x + 1, 2<x < 5 h(x) =

A: The given function is hx=ekxif 0≤x&lt;2x+1if 2≤x≤5. The function is continuums. That is,…

Q: Suppose that a continuous function g satisfies: 9(z)dr = 42 and 9(2x)dz = 12. | z9(32°)dz. Find

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Q: 2 + 7x if x < 4 Let f(x) if x 2 4. For what value of the constant c is the function f continuous on…

A: Given function is: fx=cx2+7x       if x&lt;4x3-cx          if x≥4 We know that a function is…

Q: 4. Give an example of a continuous function f : (0, 1) → R that cannot be uniformly approximated by…

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Q: 7. Suppose f is a continuous function on [a, b] and Sf(x)g(x) dx =0 for every continuous function g.…

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Q: 20. The function f is continuous at x = 1. Vx + 3 -. - V3x + 1 for x 1 x - 1 If f(x) = k for x = 1…

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Q: Suppose that a continuous function g satisfies: | = 36 and g(x)dx | 9(2x)dr = 15. 0. : = =p(= |…

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Q: Suppose that the function f: [0, 1] in R is continuous and thatf(x) &gt; 2      if 0&lt;=X &lt; 1.Is…

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Q: 4. If L is a Continuous function on H, then unique YEH such that L(x)=(x,y). уєн there is a

A: Given that L is continuous on H. To show: y is unique in H.

Q: 4. If continuous function f:[0,1] U (2,3) → R is one-to-one, is f also continuous? 1

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Q: Find a value of k making h(x) continuous on [0, 5]. kx , 0 < x < 1 h(x) = x +3, 1<x < 5

A: Given: hx=kx,0≤x&lt;1x+3,1≤x≤5 given function will be continuous over 0,5, if it will be continuous…

Q: 1. Construct a continuous function f: I R on a bounded interval I with the following property: f is…

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Q: 10 If f is a continuous function and F(x)=f(x) for all x€R, then | f(Gx)dx=?

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Q: - 35) Prove that if y=flx)is continuoUS and flx1= 0 when x is a rational number, then f Lx)=0 for…

A: The objective is to show f(x) =0 for all x belongs to R.

Q: 2x, ifx BDU G EDU y = 5, if x= 3 if x = 3 (x-5, if x 3

A: Given:   y=2xif x&lt;35if x=3x-5if x&gt;3 To prove whether y is continuous or not x=3. Solution: A…

Q: 9. (True/False) If f(x,y) is a continuous function on a closed, but unbounded set D, then f(x,y)…

A: False     because there are some functions which are continuous on a closed, but unbounded set,…

Q: 2. Consider the function f with domain (-2, 0): 3|22 – 1| if – 2<r<1 2.r2 + 2x – 4 f(x) = 3 if r = 1…

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Q: x²-1 for x# 1 - If flx) = show that flxx) is continuous at x = 1 x-1 2 for x = 1,

A: topic - limits and continuity

Q: A function ƒ is continuous for all x and has a local minimum at (2, –5). W statement below must be…

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Q: 7.14. Show that if f = u+iv is analytic in a region S and u is a constant function (i.e.,…

A: Analytic function which is a constant

Q: Suppose that  f(1) = 4, f(4) = 5, f '(1) = 6, f '(4) = 5, and f ''  is continuous. Find the value of…

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Q: 1. If g(x) is a continuous function on [2 ,9] such that , g(x)dx = 25 and ,9(x)dx = Find value of…

A: Using the properties of definite integral, it can be written as follows.

Q: Suppose that a continuous function g satisfies: | g(x)dx = 12 and g(2x)dx = 27. Find æg(3x²)dx.

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Q: Suppose that a continuous function g satisfies: g(x)dx = 42 and | g(2x)dx = 12. 2 | #g(3x²)dx. Find

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Q: 6. Determine the value of the constant a for which the function g(x) is continuous at x = 1 x* + 5x…

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Q: Prove: f(x)=5x y is continuous at x=2

A: A function fx is continuous at a point x=a, if limx→afx=fa Given function is

Q: 1. Suppose that g(x) > 0 for all r, g(x) has a continuous derivative, g(-2) = 5 8 g(4) = 3 and |…

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Q: 2. Consider the function f(x) = if x = {9 in lowest terms q > 0, p,q e Q; 0 if x e Q°,x = 0…

A: We use sequential approach of continuity to show it.

Q: Σ 1 is continuous function in R. Prove f(x) = 4 n=1 n' +x*

A: The given function is                                     f(x)=∑n=1∞1n4+x4 To show that the above…

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.