2. Suppose that f is defined on (0, b) and that f is differentiable at c € (0, b). (Do not assume that f is differentiable at any other points.) Evaluate each of the following limits; the answers will be in terms of c, f'(c), etc. f(x)-f(c) xf (c) - cf(x) a) lim b) lim x1c - C² c) lim f(x²)-f(c) x-√c

Q: Consider the piecewise defined function Find the values of a and b that make the following function…

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Q: 3. Let f(2) = if z + 2i, while f(2i) = 3+4i, is f(2) continuous at z = 2i? 2- 2i 4. Evaluate lim (az…

A: 3)Given:

Q: f(x) – f(c) If lim exists, it is called the of f at c.

A: We know, If limx→cf(x)-f(c)x-c,exists it is called the derivative of  f at c.Denoted by f'(c)and f…

Q: Let ce (a, b). If ƒ (x) is a function that is continuous at c, then lim,e- [f (x)] = lim,»c+ [f…

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Q: Answer the following True or False: If lim f(x) exists, then f(x) is a differentiable function at x…

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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: xy Let f(x, y) : and let L = lim(2,y)>(0,0) f(x, y). Which of the 8x2 + 9y2 following statements is…

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Q: 3. Describe the interval(s) on which the function is continuous. c) S(x)= x-1 2, x= 1

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Q: , if x + -1 -2, if x = -1 Prove that f(x) = {x+1 is continuous at a = -1 by showing the following.…

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Q: Let f be a function sati sfying f(x+y)= f(x)f(y)-/4-f(y) for all x, y e R. and lim f(x)= 4. discuss…

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Q: Use the definition of the derivative to evaluate the following limits.

A: Derivative of a function f(x) is defined as, f'(x)=limh→0f(x+h)-f(x)h. Derivative of a function f(x)…

Q: lim 3i 4 +

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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: Give a pair of functions f: R →R and g : R →R such that g is not tangent to f at 0 and yet lim,0…

A: To give a pair of function f : ℝ→ℝ and g : ℝ→ℝ such that g is not tangent to f at 0 and yet limx→0…

Q: If f is differentiable at xo, prove that f (xo + h) – f (xo – h) – f' (x0) - lim h-0 2h

A: Given that: f is differentiable at xo   To prove: limh→0f(x0+h)-f(x0-h)2h=f'(x0)

Q: Let f and g be functions that are continuous on [a, b] and differentiable on (a, b). Show that there…

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Q: Evaluate cot x – cos x lim z¬7/2 (T – 2æ)°

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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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Q: Consider the following graph of the function f. X-0 lim f(x) = X→0+ -3 -2 -1 4 3+ True False 2 - + 2…

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