f(x) = (x + 2x')', a= -1

Q: Determine the domain of the function and prove that it is continuous on its domain using the Laws of…

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Q: compute f ' (a) using the limit definition and find an equation of the tangent line to the graph of…

A: Plug x=4 to find f'(4) Answer: f'(a)= -1/16

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: Use the definition of continuity and the properties of limits to show that the function is…

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Q: (b) f(x) - (x + 2x')', a- -1

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Q: Let f : [1, ∞) → R be a continuous function such that lim|f(x)| = 0. For n ≥ 1, let gn : [1, ∞) →…

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Q: x? sin(1/r) if r + 0 if x = 0. if x + 0 f (x) = %3D

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Q: Evaluate the limit sin x/x as x approaches 0 when x = -1

A: We have to evaluate

Q: Use the definition of the derviative to compute the derivative of f(x) = at the 2. Evaluate the…

A: Calculate the value for f(x) at x = (2 + h) and x = 2. Simplify the numerator.…

Q: Express the limit as a definite integral on the giveninterval.

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Q: Express the limit as a definite integral. [Hint: Consider f(x) = x*.] lim n- 00 dx

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Q: Use continuity to evaluate the limit. 5 + Vx 28. lim sin(x + sin x) 5+x

A: To evaluate:   using continuity.

Q: Use continuity to evaluate the limit. lim 9 sin(x + sin(x))

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Q: Use the definition of continuity and the properties of limits to show that the function is…

A: Given:               h(t) = 2t-3t21+t3,  a = 1 Definition of continuity: A real-valued function f(x)…

Q: Write the limit as a definite integral on the interval [a, b], where c, is any point in the ith…

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Q: Let f(x) = - 3x(x − 1). Then use the LIMIT definition to find the derivative, showing your work:…

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Q: Use continuity to evaluate the limit. limit as x approaches pi sin(x+sinx)

A: Sine function, sin(x) is continuous for all the real values of x. Hence, sin(x + sinx) should be…

Q: Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form…

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Q: Use the limit definition to find the slope of the tangent line to the graph of f at the given point.…

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Q: , Use the limit definition of the derivative to prove the following special case of the Quotient…

A: Let f be a differentiable function. then its derivative is given by f'x=limh→0fx+h-fxh.

Q: Use continuity to evaluate the limit. lim , sin(tan(cos(0))) 0-37/2

A: Here we calculate the limit of given function.

Q: Determine if the polynomial or rational function is continuous at the given value of c. If the…

A: Given that: 1. f(x)=(3-x)1/5; c=2   NOTE: As per our answering guidelines, we can only answer 1…

Q: Use the definition of continuity and the properties of limits to show that the function is…

A: A function is continuous at a point if the limit of the function at that point is equal to the…

Q: Express the limit as a definite integral on the given interval. n X;* -Дх, [1, 9] lim (x;*)² + 4 n-…

A: limn→∞∑i=1nxi*xi*2+4∆x,    1, 9 Here, fx=xx2+4, a, b=1, 9

Q: 1. Express the limit as a definite integral on the given interval. Xi lim Ax, [1, 3] '(x;)² +13 n00

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Q: Use the definition of continuity and the properties of limits to show that the function is…

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Q: Prove that f(x) = lxl is continuous for all x. Hint: To prove continuity at x = 0, consider the…

A: Consider the given function as fx=x Use absolute property x=-x        x<0x        x>0

Q: Use the limit definition to compute f '(a) and find an equation of the tangent line. f (x) = 2x2 +…

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Q: Use the definition of continuity and the properties of limits to show that the function is…

A: Given,           ht=2t-3t21+t3 , a=1

Q: Find the limit and determine if the function is continuous at the point being approached. (- lim sin…

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Q: compute f'(a) using the limit definition and find an equation of the tangent line to the graph of f…

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Q: Use the definition of continuity and the properties of limits to show that the function is…

A: A function is continuous in an interval if is continuous on every point of the interval. Now, a…

Q: Now consider the equation for the function above: ex+2 + 4, Vx + 3 + 1, х 2 f(x) 2, 3x, Use the…

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Q: Given f(x) = |2x – 2|. 1) At which points is f not differentiable? Prove your claim using the limit…

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Q: 11. ((x) – (x + 2r'), a--1

A: To find the limit.

Q: Use the definition for continuity to determine whether the function continuous at x= 1 and x= 2?

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Q: Assume f is continuous on an interval containing zero anddifferentiable for all x not equal to  0.…

A: Consider the Darboux Theorem: Let I be an open interval, f:I→ℝ, a real- valued differential…

Q: Comment on the continuity of f(x) at x = -2, x = 0 and x = 3 If discontinuous, what is the type of…

A: At x=-2, L.H.L.=limx→-2-fx=2R.H.L.=limx→-2+fx=-1f-2=2 Since L.H.L.≠R.H.L., therefore the limit does…

Q: Determine the limit as x approaches the given x-coordinate and the continuity of the function at…

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Q: Use the limit definition to find the slope of the tangent line to the curve f(x) = square root of…

A: The given curve is  f(x)= y = √(x + 2).. (1)

Q: 25. Let f(x) be a continuous function. Express + ()] lim п—оо as a definite integral.

A: Definite integral in the form of the limit of the Riemann sum:   Let f(x) is a continuous function…

Q: Use the limit definition to find the slope of the tangent line to the graph of f at the given point.…

A: The slope of the tangent to the graph 13−x2 by the limit definition.

Q: Use the limit definition to compute f '(a) and find an equation of the tangent line. f (x) = x−1, a…

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Q: (2) Determine if each function is continuous at th limits and the definition of continuity. If any…

A: for a function to be continuous it is compulsory that the right-hand limit exists and is equal to…

Q: Express the limit as a definite integral on the given interval. lim x, In(2 + x^)Ax, [3, 4] n- 00

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Q: Use the limit definition to find the derivative of f(x) = 5x? + 2x – 7 at x = -2. Show all 1. :…

A: Consider the provided function, fx=5x2+2x-7, x=-2 find the derivative by using the limit definition.…

Q: Now consider the equation for the function above: 2x + 3, x 4 - x-3 ’ 2, x = 4 Use the equation and…

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Q: Use continuity to evaluate the following limit lim sin(x + cos(x)). Limit:

A: Given; Limx-π Sin(x+ cos(x)).

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.