2. Let f: DR where DCR is open, and suppose that f € C²(D). Let xo E D and suppose that B(xo, e) CD for some > 0 (this is guaranteed by the fact that D is open). (a) Prove that 1 8² f (xo) = lim lim (f(xo + tek + sej) — f(xo + tek) — f(xo + sej) + f(xo) Әхкдх ј + f(x))] t-0 8-0 ts and that 8² f дхјдхк 1 -(xo) = lim lim (f(xo + sej + tek) — ƒ (xo + se;) − ƒ (xo + ter) + f(xo); -f(xo))]. 8-0 t-0 ts

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: evaluate the limit

A: Given that: The limit is limx→0 1 - 2x1x .

Q: 3. Use the Squeeze theorem to prove that lim x- sin () = 0. Explain thoroughly. x0 (Hint: Use f(x) =…

A:

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Q: Use the Squeeze Theorem to find limx→c f(x).

A: Given that 4-x2≤f(x)≤4+x2 To find the limit of the function using the Squeeze theorem when x→c=0.

Q: (2) Use the definition of limit of a function to find lim x³-3x-2 x+2x2 - 3x+2

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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…

A:

Q: f(r) where f(r) and g(r) are polynomials. g(r)' 2. Describe a strategy for solving limits of the…

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: Let a and b be constants in R. and If lim (f(x) - (ax + b)) = 0, what is the value of b? x→→∞0 b= =…

A: Solution to find the value of b  of this limiting problem is as follow.

Q: 4. Let f RR and let cЄ R. Show that limf(x) = L if and only if lim f(x + c) = L. x-e x-0 PLEAEhelp…

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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: Let f(x) = a² + 3. Find lim * + h) – f(x) h→0 h Select one: O a. 0 O b. 3 O c. 2x+3 O d. 2x

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Q: Show that lim e/n = 1 for any constant c> 1. Use the squeeze theorem (or any other method) to…

A:

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

A:

Q: I if X > 4 and J (X). 4X + 1 i1 X<4. Find lim f(x) x→4+

A: Given; f(x)=√x-1 if x&gt;4 and f(x)=x2-4x+1 if x&lt;4.

Q: 3. If f is a continuous function, what can you say about lim f(x)?

A: Given thatf is continuous To determinelimx→1f(x)

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

A: Above question is solved.

Q: suppose f,g an d h are functions which g(x)<=f(x) <=h(x) ,for all x in an open interval containing a…

A: No. For the given continuous open interval domain, the limit of f(x) does not exist.

Q: Calculate the limit limx→1sin(x−1)x4−1.

A: value of sin(x) at x=0 is: sin(0) = 0

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

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A:

Q: 5. Calculate lim f(x), where x+1+ f(x) = = its 12.05 (c)-2 (d) 2 (e) does not exist 5A 5B if x 1 x-1…

A:

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A: Use the chain rule of differentiation and then solve .   As per the bartleby guidelines I am allowed…