Given the conditions below, a function f is said to be continuous at "a" if (1) f(d) exists; (2) lim f(x) exists; and (3) lim f(x) = f(d).

Q: Question 8 If f(x) is a constant function. Then for any real number a f(a + h) – f(a) lim h 0 =0. h…

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Q: f(a) exists lim, a f(x) exists the two above values are equal: f(a) = limr→a ƒ(x) x²–4x+3 if x 3…

A: Topic -Limit and continuity

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: Let f be a function that satisfies |f(x)|<M_for all x# 0. Show that lim(x f(x))=0.

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

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Q: Let f(x) be defined on an interval (a,b) and suppose that f(c) #0 at some c where f(x) is…

A: 1)Therefore correct option is4) All of the above

Q: Prove positive function so that lim + (x)=16, then lim√(x)=4 directly from definition that it f is a…

A: Given that Let  f is a positive function so that  limx→-4f(x)=16  .

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: 2x+1 Consider the function f(x) Determine lim f(x).

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Q: Evaluate f(a), lim f(x), lim f(x), lim f(x). Is f(x) continuous at x = a. x→a- x→a+ xa if x 4

A: For 4^- we use the first piece of the function

Q: If f: Z → R is a function, then the expression lim f(x) is undefined. x-2 True False

A: If f:ℤ→ℝ, is a function then  limx→2f(x) is undefined. To Check: Whether the given statement is true…

Q: x+3,if x2

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Q: Sketch the graph of an example of a function f that satisfies all of the given conditions.  lim x→0…

A: Answer

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.