(1) The parts in this question are not related to each other. 4 1 (a) ) Using the e – ổ definition of limit, prove that lim 1+3 3r – 1 (b) Let f : R →R be a function with the following property: \f(a) – f(b)| < 4|a – b| for all a, b e R. Prove that f is continuous everywhere. (c) Show that the function g(x) = r³ – 7x² + 25x + 8 has exactly one real root.

Q: Define the function f(x) to be 1/x for x for all x that are (2n,2n+1) for even n and -1/x for x for…

A: The given function is

Q: 6) Let f be a function with domain R. Assume f is increasing and bounded above. Let M be the…

A:

Q: Suppose thatƒ and g are functions such that lim f(x) = 9 x →4 lim g(x) = 4 Evaluate the following…

A:

Q: f(x): = ax² + x - b, ax + b, 2ax - 7, (a) Find the values of a and b such that f is continuous on…

A:

Q: (a) Using only the e,6-definition of limit, prove that 1 lim = 1. x--1 x2 (b) Let n be a fixed…

A:

Q: Graph the function (1 x<-1 -x -1<x< 0 f(x) = {1 x = 0 0 <x <1 1 x 21 Then discuss, in detail,…

A: Given: f(x)=1,x≤-1-x,-1&lt;x&lt;01,x=0-x,0&lt;x&lt;11,x≥1 To Discuss: Limits, one-sided limits,…

Q: Let f(x) be defined on an interval (a,b) and suppose that f(c) #0 at some c where f(x) is…

A: 1)Therefore correct option is4) All of the above

Q: If 2x 0, use the Squeeze Theorem to find lim g(x).

A: Given that : as per squeeze theorem

Q: Give examples of functions f and g such that f and g do not have limits at a point c, but such that…

A: The objective of this question is to find two functions, f and g, that do not have limits at a…

Q: (a) Using the formal definition of a limit, prove that f(x) = 2x³ - 1 is continuous at the point x =…

A: The given function is                      f(x)=2x3-1 Definition of limit:  Let f(x) be a function…

Q: Question: Prove that the limit does not exist. -2|3x| lim 3x

A:

Q: for all a E R \ {0}. Using the e-ð definition of the limit of a function, prove that lim Do not use…

A:

Q: Which of the following statements about limits is always true for any function f, and a, k E R? lim…

A:

Q: Determine the following limits from the graph of f given to the right: a.) x→ -1* lim f(x) p.) lim…

A: According to the graph    Function at x=-1 is not defined, according to the graph because a hollow…

Q: g(x) (4) Let g be a function such that lim to write proofs from the definition of the limit. Make…

A:

Q: Which of these is not a condition in determining the continuity of a function f(x)? Select the…

A: