3 - x, x<2 Q.No.15: Is f(x) = {+2 x22 continuous at every x € R. x+1 xs1 -.

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

A:

Q: 1. Supposef is a continuous function on x 2 0,f'exists for x> 0, f(0) = 0 and f'is monotonically…

A:

Q: Find the x-values (if any) at which f (x)= 1s not continuous. x²+ 3x O fx) is not continuous at x =…

A: It is known that to check the continuity, we can equate the denominator part with 0.   Here, we…

Q: (0,1). Prove the following functions is uniformly continuous on sin x f(x) =

A: Uniformly Continuous:-  Here we prove the result using a theorem of uniformly continuous.

Q: 2. cuts the curve at Q, such that Let ƒ be a continuous and differentiable Tunction in (x,, x.). If…

A:

Q: 4- If a function is differentiable at zo, then it is : a) Differentiable on C b) Continuous on C c)…

A: To answer the given question about the complex function.

Q: 5. If g is continuous on an interval [a, b] and has an extremum at a number c € (a, b), then the…

A: I give two examples for this statement.

Q: 1 x2 - х х x 2 Vx -2, Identify if the function g(x) is continuous/discontinuous at x=1. If…

A: Given: f(x)=1x2-x+1xx&lt;0-10≤x&lt;101≤x&lt;21x=2x-2x&gt;2 To determine: The given function is…

Q: 2. Suppose f20, f is continuous on [a, b], and f(x) dx = 0. Prove that f(x) =0 for all x e [a, b).…

A:

Q: Find all values of c such that f is continuous on (-∞, ∞). S 4– x², x c с 3 (smaller value) C =…

A:

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: 3 Suppose that e(x) and f(x) are differentiable on [0, 1] and that 0 < |e'(x)| < |f'(x)| for all x €…

A:

Q: Suppose that f is a continuous function for which f(1)÷0 and f(t) dt=0. What can we conclude about…

A:

Q: If f is continuous on [-1, 1] and f(x)dx = 0, then there exists some ce [-1,1] for which f(c) = 0.

A: Given: f is a continuous function on the interval -1, 1 and ∫-11f(x) dx=0. Prove: There exists some…

Q: Sketch the graph of a continuous function on [0,4] satisfying the given properties. f'(x) = 0 for x…

A: f'(x)=0 for x=1,2 and 3

Q: Find all values of c such that f is continuous on (-0, 0). 1- x2, f(z) = { – x², x. C = (smaller…

A: Given function is, fx=1-x2   , x≤cx          ,x&gt;c  Since 1-x2 and x are continuous on the…

Q: 3. f(x) = Vx - 5 is continuous on the interval [5, 0) %3D

A:

Q: (x²-x for x +0 If f(x) : and if fis continuous at x = 0, then k= 2x f(0) = k -1/2 O 1/2 O 1

A: f(x) = (x2 - x)/2x limx→0 x2-x2xlimx→0 x(x-1)2xlimx→0 x-12= -12

Q: 1 Let f(x) (x – 1)(x – 4) Select one: a. fis continuous on (1, 4) O b. fis continuous on [0, 3] O c.…

A:

Q: The function g is continuous on the interval [a, b] and is differentiable on (a, b).Suppose that…

A: g(x) = 0 for 7 distinct values of x in (a, b).Drawn a rough sketch of the case

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

A:

Q: (4.)Consider the function f continuous on [-1,1], such that f(0) = 0, and f'(z) = { 3x2 for -1 < x <…

A: Explanation of the answer is as follows

Q: 2. If f is continuous on [0, 1] such that x* f(x) dx=0, for n=0, 1, 2, 3,.. then show that f(x)3D0…

A:

Q: 3. Find the absolute maximum and the absolute minimum of f(x) = x-2 tanx on the interval [0,4].

A:

Q: 1-cos 2kx 3. Find k such that f (x) = x2 is continuous everywhere. 4 - 7k X = 0

A:

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

A:

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

A:

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

A:

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…

A:

Q: ... uavarasvo vr-vv, find the x-values (if any) at which f is not continuous. Which of the…

A:

Q: f(x)= X-I x=1 √x-1 X = 1 1123 -s continuous at x = 1?

A:

Q: for z+ - and Let f(z) = for z or (a) Show that is continuous at -n/2.

A: a) When x≠-π2 limx→-π2cosxx2-π24

Q: Suppose f(x) is continuous on [0, 1], with f(0) = 2 and f(1) = 0. Then there must be a value x in…

A: Here given that f(x) is continues.

Q: The graph of the continuous function g(x) has select values displayed in the table. -2 -1 1 4 0. -6…

A: Critical points are points where function is not defined. function derivative is zero derivative of…

Q: 1. Does there exist a differentiable function f that is continuous on the closed interval [1, 4],…

A: Mean Value Theorem (MVT):   If f(x) is continuous on the closed interval [a, b] and differentiable…

Q: S [[x+1]], Determine iF f(x) = if x ≤ -1 2 tanh (H), ip-IxO (x+1), < <) In [cash (x+1) + sinh…

A:

Q: II. Determine if [x+1]. if x < -1 S x + 1 f(x) = 2 tanh (x+¹) if 1<x< 1 1-x In [cosh (x + 1) + sinh…

A: It is solved whether f(x) is continuous or not.

Q: 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…

A: The given functiong(x)=7x+2-6x+4x-2,if x≥-27 and x≠2k,if x=2

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

A: Givenh(x)=ekx    0≤x&lt;2          x+1  2≤x≤5To find a value k when h is continuous on…

Q: 1-x², x 2 at x = 2.

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

Q: x-1 for x 1 x -1 show that flx) is continuous at x = 1 !3! for x 1,

A:

Q: 2. The function (x) = x² is differentiable at x = 1. Find the function & such that o(x) = o(1) +…

A: Introduction: A technique for determining a function's derivative is differentiation. Mathematicians…

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…

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…

Q: 18.Q. Is it true that function on R? to R is continuous at a point if and only if it is both upper…

A: exercise 18.Q

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.