Suppose that lim, →o(f(x) – 9(x)) = 0 and h(x) is a function with domain R. a) Assume that |h(x) – h(y)| < 3|x - y| for any x, y E R, if lim,→o(h(f(x)) – h(g(x))) = 0 is always true, provide a proof; if not, provide a counterexample.

Q: If f(x) is defined by the following ²-16 =x2-4 2-3x-1 Let f(x) x> 2 *<2 Then find the lim f(x) & lim…

A:

Q: 3) Consider the function f(x) = VI+: a) Write its domain. Find lim,→0+ f(x). b) Find intervals of…

A: The function, fx=x+1xa To write its domain. To find limx→0+ fx.b To find the intervals of increase…

Q: Re Prove directly from definition that it f is a positive function so that lim +(x)=16, then…

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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…

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Q: Suppose f: (0, ∞o) → (0, ∞o) is a continuous function. True or false: lim √f(x) = = /lim f(x). 2-5…

A: We have to find whether given statement is true or not?

Q: Sketch the graph of a function, f(x), that has each of the following properties: • Domain: x €…

A: Given some properties of function to draw a graph of function f(x)

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: If f(x+ y)= f(x)f (y) for all x, y eR and f (x)=1+g(x)G(x), where lim g (x)=0 and lim G (x)…

A: The problem has been solved in detail in the next step.

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: Let f(x) = |x|/x . Prove that limx→x0 f(x) does not exist by using a sequential argument.

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

A: Above question is solved.

Q: If the ratio of boys to girls in tennis camp is 3:5 if there are 64 campers, how many are girls 

A: We find out how many girls as follows:  • We first need to determine the total number of parts in…

Q: Compute lim,0f (x) for each of the following functions f (x). 3x (c) f (x) = Vx2 + 8 3x (d) f (x) =…

A: topic - limits

Q: Q3 (a) State Rolle's Theorem. i) Verify Rolle's theorem for the function f(x) = (x – a)2020 (x –…

A: Since you post more than four subparts questions as per our guideline we can solve only first four…

Q: Evaluate f(a), lim f(x), lim f(x), lim f(x). Is f(x) continuous at x = a. x→a- x→a+ xa if x 4

A: For 4^- we use the first piece of the function

Q: Prove that v5 is irrational.

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Q: Let f and g be real-valued functions defined on (b, ∞). Suppose that limx->∞f(x)=L and…

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Q: x² – y + 1 f (x, y) x² – y? + 1 -

A:

Q: If f(x) = 2x? – x + 2 , what is lim 2x2 – x + 2, lim 2x2 – x + 2, andf(2)? Is f(x)continuous at 2? O…

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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.