Theorem 3.2.10. Let ICR be an open interval, let e E1, let f, g:1-{c}-R be functions and let keR. Suppose that lim f(x) and limg(x) exist. 1. lim [f+g](x) exists and lim[f+g](x)= lim f(x) + limg(x). 2. lim f-g](x) exists and lim[f-g](x)= lim f(x)- limg(x) 3. lim kf1(x) exists and lim k f1(x) = klim f(x). 4. lim [fg](x) exists and lim[fg](x) = [lim f(x)] · [limg(x)}. lim f) 5. If limg(x) # 0, then lim [4] (x) exists and lim [4] (x) = Prove above theorem. lim()

Q: If the fucntion f is undefined at x then lim f(x) does not exist. = c, C, FALSE O TRUE

A: Given- If the function fis undefined at x=c then limx→cfx does not exist. To Check- If the above…

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: 7. Use the definition of the limit of a function to show that if lim f(x) = a and lim g(x) = b, x→x0…

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Q: 1. Sketch the graph of a function f that satisfies all of the given conditions. Label your graph…

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Q: 5. Sketch the graph of one function y = f(x) that satisfies all of the requir- ments below. lim f(x)…

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Q: 5. The graph of f (x) is shown below. y (a) lim f(x)= X-1 (b)lim f (x)= (c)lim f(x)= X-1

A: Given Graph of fx

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: 4. Is it possible that there exists some function g(x) and some constant L such that lim g(x) = lim…

A: First we need to understand definition of limit:

Q: n each part below, give an example of a function which satisfies the given criteria. Justify that…

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Q: Given the graph of the function f below, Which of the following must be true? 3 2.

A: Graph is given

Q: Let f and g be real functions. Show directly from the definitions that if f is everywhere continuous…

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Q: 7. Identify if the statement is Always True, Sometimes True or Never True. If lim f(x) exists, then…

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Q: Show that if lim f(x) = lim h(x) = 3, and if f(x) ≤ g(x) ≤h(x) for 2-0 x-0 all z € (-1,1), then lim…

A: Given that limx→0fx=limx→0hx=3 and fx≤gx≤hx for all x∈−1,1. We have to prove that limx→0gx=3.…

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.…

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Q: Given the function f(x) = Which f the following is/are true? (x- 1)(x + 1) 1. lim f(x) = +00 2-1+ 2.…

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Q: lim F(x),

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Q: 1) Use the graph of f below to estimate the following: m 0 -1+ % 4 a) lim f(x): x-1 b) lim f(x)= c)…

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Q: 3. Find the limit of +2x+3 lim X-2 V x+5

A: As per Bartleby guidelines, we will solve only first question To find the limit of limx→2x2+2x+3x2+5

Q: 3. Sketch a graph of a function (x) for w ich all of the fo lowing are true: lim-2' f(x) = -4,…

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Q: 5. Determine the end behavior of the given functions. lim f(x)= lim f(x)= 2-90

A: We can see clearly   from the graph       as   x is approaches   to  toward   2 from RHS     the…