7. Let f,g: R → R, and c Є R. 1. Prove: if both lim f(x) and lim(f + g)(x) exist in R, then lim g(x) exists in R. х-с x c x c 2. If lim f(x) and lim(fg)(x) exist in R, is that true lim g(x) exists in R? Show your x c argument. x c х-с

Q: (b) Assume that lim,6 f(r) = 9 and that lim,6 g(r) = 3. Compute lim f(r)g(r) - f (x)f(x) + g(x)g(x)

A: To Determine: The given expression:

Q: |x+1| if x + -1 13. Consider the function f(x) = {x² - 1 Evaluate lim f(x). if x = -1 A) - C) DNE B)…

A:

Q: (4) Let f(x) be the function whose domain is all of R given by the rule 42 if r € Z and f(x) = 10…

A:

Q: Suppose (a, b) is on the boundary of the domain of f. Explain how you would determine whether lim…

A: Given : (a, b) is on the boundary of the domain of f

Q: ) State the definition of the "limit of a function": The definition begins: GIVEN: f:(x1, a)U(a, x2)…

A:

Q: 1 if x<1 1. Consider the function f(x)= . Evaluate lim f(x). if x 21 2х - 3 C) 0 D) 1 A) 2 В) -1

A: Here we have to use the concept of limit.

Q: Let the function f: DCR R be defined by f(x) = lim n0 2+ " (a) Evaluate lim,+-1- f(x) and lim,-1+…

A: The given function is  fx=limn→∞xn2+xn

Q: Exercise 2. Let f: (1, 00) → R and g : (0, 1) → R be such that g(x) = f(1/x). Prove that lim f(x) =…

A:

Q: Evaluate lim a,2)→(2,3) 2y2 +3xy 5xy+a?

A: This question is related to limit of function of two variables, we will use direct substitution…

Q: Use the graph of f(z) below to find lim f(z) 10 y -4 -2 10 a) O-3 b) O2 c) O-4 d) O-1 e) 00 f) O…

A:

Q: State the definition of the "limit of a function": The definition begins: VEN: f:(x1,a)U(a, x:) – R…

A: The definition for limit of a function: The number L is called the limit of a function f(x) as x→a…

Q: Let f,g: RR, and c E R. 1. Prove: if both lim f(x) and lim(f + g)(x) exist, then lim g(x) exists. x…

A: Let's solve each part of the problem based on the given mathematical scenario.1. Prove: if both…

Q: USE SQUEEZE Theorem do poho Lim I'm x² esin(x) и х 0+X

A:

Q: x→4 5. Sketch a function f such that: lim f(x), lim f(x), and f(4) all exist, but no two are equal.…

A:

Q: -4 Determine the limit: lim -i (и — 1)2

A:

Q: b) Evaluate lim g(x) if 2x − 1 ≤ g(x) ≤ x² − 2x + 3 x→2

A: Above question is solved.

Q: prout Lim An = Lim sup An = kzu

A:

Q: Evaluate the limit t³ - 1 1 lim t 1 ²

A:

Q: Consider the funetion 1) Find the domain of glx) 2) Find lim g(x) and limg(x). 3) Find oblique…

A: Solution: Given data ,  g(x)= 2√(x2+3x)

Q: Use the Squeeze Theorem to show that lim x2 co X-0 cos(207TX) = 0. Illustrate by graphing the…

A:

Q: Evaluate: Lim

A:

Q: 40. Show that if both lim f(x) g(x) and lim g(x) cxist and f(x) g(x) 8(x) lim g(x) # 0, then lim…

A: Given that limx→c fx gx   and  limx→c gx both exist and limx→c gx≠0. This implies that limx→c fx gx…

Q: x² - 1 if x 2 x→2* а) 4 b) – 3 с) 3 d) – 4

A: Given function is  g(x) = x2-1   if x≤2x+2    if  x &gt;2 the function is defined in two intervals.…