Definition 3.2.1. Let /CR be an open interval, let c € 1, let f: 1- {c} → R be a function and let L E R. The number L is the limit of f as x goes to c, written lim f(x) = L., if for each ɛ > 0, there is some 8 > 0 such that x €I- {c} and |x – e| < d imply |f(x) – L| < ɛ. If lim f(x) = L, we also say that f converges to L as x goes to c. If f converges to some real number as x goes to c, we say that lim f(x) exists. A Definition 3.3.1. Let ACR be a set, and let f:A→R be a function. 1. Let cE A. The function f is continuous at c if for each ɛ > 0, there is some 8 > O such that x E A and ļx– c| < 8 imply |f(x) –f(c) < ɛ. The function f is discontinuous at c if f is not continuous at c, in that case we also say that f has a discontinuity at c. 2. The function f is continuous if it is continuous at every number in A. The function f is discontinuous if it is not continuous.

Q: Using the € - d definition of limit, prove that 2x lim x 2 x + 6 1 NI

A:

Q: Describe about the Why the Central Limit Theorem Works.

A:

Q: how do you proof by the limit definition that the limit of x to 1 of the function…

A: Ans-We are given function

Q: Explain the Central Limit theorem with proposals

A: We want to define central limit theorem

Q: lim Vx +x+1 x- - X-∞

A:

Q: 3+x; xL1 3- X 14 X 8x 40 x 480 11x -4X; 80LX 280

A:

Q: If f is a function then lim f(z,3) = f(2, 5). (29)H2,5) True False

A:

Q: Evaluate the limit of the function below x² – 4| lim, 'x→2 х - 2 O The limit does not exist O 1 O 2

A: limx→2x2-4x-2consider the function x=x,x≥0-x,x&lt;0

Q: Prove that limit x 0+  sqrt x =0

A: To Evaluate: limx→0+x.

Q: 4 – x lim x - 4 5- Vx² + 9

A:

Q: 1/x a* + b* +c* Evaluate the following limits (i) lim 3

A:

Q: Create an Central Limit Theorem Daily Life Own Word Problem and Cite the importance of using it.

A: First of all, the central limit theorem states that the distribution of sample means approximates a…

Q: G Eulah the but [e-)(3-2:)41-3,] lim Z-> 2ti and use the the limit

A:

Q: A uniform limit of continuous functions is continuous.

A:

Q: write on history of limit of functions and its importance of in other science?

A: The limit of a function is a fundamental concept in calculus and analysis. It describes the behavior…

Q: lim 1418 r = ∞ zª

A:

Q: Select which of the four statements is/are TRUE? One or more options can be correct. fla + h)-f(a) A…

A:

Q: Find a basis for the subspace of R³ spanned by S. S= {(4,2,-1), (1,2,8), (0,1,2)}

A:

Q: 1 1 lim X/1+ x

A: The given limit,       limx→01x1+x-1x

Q: Find the limit

A: limx→∞2x-1x+1