Let limx→a f(x) = ϕ and limx→a g(x) = γ. Then the following statements apply: a. limx→a (f(x) + g(x)) = ϕ + γ b. limx→a (f(x) − g(x)) = ϕ − γ c. limx→a (f(x) g(x)) = ϕ γ d. limx→a f(x)/g(x)=ϕ/γ provided γ ≠ 0 e. limx→a f(g(x)) = limx→γ f(x). Now I have the question , it is required to prove the statements only using the theorem given above: Take f, a and ϕ as in theorem. Show with the calculation rules that: a. for every k ∈ R holds limx→a kf(x) = kϕ; b. if P(x) is a polynomial is then limx→a P(x) = P(a); c. for every k ∈ Q holds limx→a(f(x))k = ϕk, provided k > 0.

Q: If V 16x <f(x)sx+4 for all positive x-values, then what is lim f(x)?

A:

Q: 3 Use the Squeeze Theorem to evaluate lim f(x), where f(x)=x²cos (-). x A B C D - 0 0 1 8 -0-x

A:

Q: Given that f(x) = 2x² +1, find:“ f(x+h)- f(x) f(x+h)– f(x). lim h→0 h h

A:

Q: f(x) : II. Determine the limit of the function, g(x) = x² + 2, given its table of values. -3 11 -2…

A: Given :                              g(x) = x2 + 2         (1).…

Q: Let f(x) = x3+x. Evaluate lim s→0 (f(x + s) − f(x))/s .

A:

Q: Use the given graph to: + + + -4, -3 -2 4 3 0 1 2 3 4 X Find lim f(x). X→

A:

Q: If f and g are continuous functions with f(3) = 5 and lim,- [2f(x) – g(x)] = 4, find g(3).

A: Use the fact that limx→cfx+gx=limx→cfx+limx→cgx. Also, limx→ccfx=climx→cfx Now,…

Q: If 4x − 1 ≤ f(x) ≤ x² − 1, find lim f(x). x-4

A: Limit

Q: If lim f(x) = 4, what can you say about ƒ(1)? x-1

A:

Q: Given the three basic functions: /(x) = sin x f(x) = log* f(x) = 2^* Determine how and which of the…

A:

Q: Explain, using limit definition, why f′(x) = 0 for any constant function f(x) = c.

A:

Q: Find the limits. F(x) = 2x? - 3x + 4, g(x) = Vx+ 2 (a) lim f(x) x-2 (b) 9(x) lim g(f(x)) = X-2

A: We have to find the limit

Q: Let f(x) = 2x2 +1. Evaluate f(4 + h) - f(4) lim h-0 h

A: Given, The function f(x)=2x2+1.

Q: If f(x) = Vx² + x – x, find lim f (x), where f(x) is defined at X → 0.

A:

Q: Take f, a and ϕ as in theorem. Show with the calculation rules that: a. for every k ∈ R holds limx→a…

A: Let lim x→a f(x) = ϕ and lim x→a g(x) = γ. Then the following statements apply:a. lim x→a(f(x) +…

Q: If 3x − 1 ≤ f(x) ≤ x2 − 3x + 8 for x ≥ 0, find lim x→3 f(x).

A: Here we use sandwich theorem Also known as squeeze theoremThis theorem states that , if gx≤fx≤hx in…

Q: Justify your answer. f(x) Is there a function f such that lim, 1 f(x) = 0 and lim, 1 = -∞?

A:

Q: -2-1 1 2 3 4 5 -2+

A: We have to find the limit limx→4fx

Q: 1 if x-1 4. lim x→- 1 5. x f(x) f(x)

A:

Q: Let f(z)=z2 - 7z+10 the find lim(zk )f(z)=?

A: