Limits of Average Rates of Change Because of their connection with secant lines, tangents, and instanta- neous rates, limits of the form f(x + h) – f(x) lim h→0 h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f. 57. f(x) = x², x = 1 59. f(x) = 3x – 4, x = 2 61. f(x) = Vx, x = 7

Q: Use the following definition of a derivative fla=lim f\x-f(a) x-- a x-a To find fa), where f x=x°-…

A: Given, f(x)=x3-2x  To find f'(a).  Solution: f'(a)=limx→af(x)-f(a)x-a      =f(a)-f(a)a-a=00

Q: Question Let k(æ) T@1g#) if f(x) = 3x +2, g(æ) = 3x – 4, and h(x) = -a² – 2x +4, what is k' (1)?…

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Q: Sketch a graph of a function f(x) that has the following properties: lim as x approaches -infinity…

A: f'(x) > 0  the function is increasing f'(x) < 0  the function is decreasing f''(x) > 0  the…

Q: On graph (A) draw a function that is continuous on [-2, 2]. On graph (B) draw a function that has an…

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Q: 3. For the function shown below, determine the following: a. limf(x) b. limf(x) c. lim f(x) AAY 3 21…

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Q: 72-1402 40+2 o f (x)= [0]. Let f (x) = The lim, ∞

A: We will take x^5 common from numerator and denominator.

Q: use the definition of the derivative, lim h-->0 (f+h)-f(x)/h, to find the following: f'(x) if f(x) =…

A: Given:f(x) = -2x+7to find:derivative of f(x) by limit definition.f'(x) = limh→0 f(x+h)-f(x)h

Q: Determine whether f(z) = zRe(z) is differentiable using the definition f'(z) = f(z+Az)-f(z) lim Az-0…

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Q: Chapter 2.1 (The limit definition of derivative needs to be used Find the derivative of f(x)=z² – 5z

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Q: In Exercises 41–44, sketch a possible graph for a function f that hasthe stated properties.

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Q: For the function f graphed as follows, approximate each of the limits and values in Exercises 31–34:…

A: 31. lim  x→-2-f(x) = -2

Q: Status. Fill in the blanks below to find f'(x) when f(x)=x²+3. Note: to type a function like x2,…

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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.