a. If f is continuous, then ſ f(x) dx = lim ſ´, f(x) dx. t-00 b. If f(x) < g(x) and g(x) dx diverges, then ," f (x) dx also diverges. x²-4 B с. x(x2+4) can be put in the form4+ x2+4 d. If f f(x) dx and f* g(x) dx are both convergent, then S"If (x) + g(x)] dx is convergent.

Q: 4 if x 2 Find the value of c such that lim f(x) exists. x→2

A: Given the piecewise function :

Q: Show that (2o + h)" – 2ö = nz- | n-1 lim h→0 h for any zo E C.

A: To show: limh→0z0+hn-z0nh=nz0n-1for any z0∈ℂ

Q: 21. If f(x) = In x, then lim f (x) – f(3) is x→3 x – 3 1 (A) (B) e (C) In 3 (D) nonexistent

A: topic - limits

Q: 2.15 Suppose there exists a E R such that x > a implies f(x) > 0. Prove: 1 lim f(x) = 0 + lim 0. %3D…

A: We have to prove the given limit.

Q: 5 Suppose that f: A -R is a function defined on an open interval A, and a E A. (a) Show that if r…

A:

Q: - 6x - 15 if x -3 Calculate the following limits. Enter "DNE" if the lim f(x) = I -3

A:

Q: Let f(x) be given by 9x - 27 x 3 where K is an unknown constant. What value should K be replaced…

A:

Q: In (x +h) – In (x) is When x = 2e, lim h→0 h

A: Substitute x=  2e above

Q: :f and the functions fn are all monotone increas- If limn0. fn ing, must f be monotone increasing?…

A:

Q: Find the value of the limit below. [4 – 6(x + Ax)]? – (4 – 6x)² - lim Ax → 0 Дх Specify the function…

A:

Q: If f(x) is differentiable at x = 2 and lim2 x*f(2)-24 28, then f'(2) = %3D x-2 Select one: O a. -1 O…

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Q: What is (ro s)(x), if r(x) = V2x – 4 and s(x) = x + 2x? O a. V2x- + 4x – 4 | O b. V2x2 + 4x - 4 O…

A:

Q: Let f and g be functions defined as f(x)=x²- 2x + 4 and 25 4 DNE 1 g(x) = 3x² + ²3. If h(x) ≥ g(x)…

A:

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

A:

Q: Suppose f(x) and g(x) are continuous functions, and | f(x) g(x) if x > 0, h(x) : if x < 0 . if x = 0…

A: Given query is to find the wrong statement.

Q: For f(x) = 3x*, find using the definition, dx dy Ay f(x + Ax) - f(x) lim Ax-0 Ax lim dx Ar-0 Ax (a)…

A: Given:- f(x) =3x4  To find:-  Using definition of derivative find dy/dx

Q: 12. Suppose f € C[0, 1]. Prove that lim | f(a" )da = f(0). n>00

A:

Q: > If f is a continuous, decreasing function over [1,0) and the lim f (x)=0, X00 then :)dx is…

A:

Q: 4. The following functions are discontinuous except? Зх +5 5x +1 3x А. В. cScx С. D. x² - 8x +12 x?…

A: As per the company guidelines we can solve first question. I hope that the given solution will help…

Q: 1. For which function below does lim f(x) = -∞? X-2+ 1 f(x) = - x-2 1 f(x) = - (2 – x) x² - 4 a. с.…

A: As per our guideline we are supposed to be answer only one question you can repost other questions…

Q: - If z -1, 1x 0 and f is function such that Inr < f(z) < then lim f(z) = 1+1 A B C 1 D +x

A: on applying squeeze theorem we get limit of f(x) =0

Q: 1. Show that for all f e L'(R) lim %3D

A: Put the delta = 1 in integral function

Q: lim f(x) = 81 (a) VF(x) (b) [3f(x)] (c) [f{x)]²

A: Using rules of limits.

Q: f is odd, J'(x) |for æ > 2, f"(x) > 0 for 0 3, lim f(x) = -2

A: First we will construct the graph for x&gt;0.  Now, point x=2 is minima and x=3 is inflection point:…

Q: b. If f(1) is undefined then f (x ) has a vertical asymptote at x = 1. lim f(x) c. If exists, then…

A: As per our guidelines we are supposed to answer only first three parts. Kindly repost others part as…

Q: 2. Write the limit definition of f '(a) and use the definition to find the indicated derivatives.…

A: The objective is to write the limit definition of f'(a) and use the definition to find the indicated…

Q: e n Show that for any &> 0 and S, ₁X₁, lim P(S₁-0 ≥ 8) = 0. =

A: We will use Chebychev's inequality to find the required limit of probability.

Q: 1. Given the function F(x)= 3x-7, x < 4 (x² - 2x, x ≥ 4 I. lim F(x) exists. X-4 (mumixat endorf a)…

A:

Q: if x 2 7 x2 - If f (x) I→2 exists. Enter DNE if the limit does not exist.

A: limx→a f(x) exists if and only if limx→a+ f(x) = limx→a- f(x) exists and same.

Q: 6x – 6 if a 4 Show that f(x) has a jump discontinuity at a = 4 by calculating the limits from the…

A: Let fx=6x-6if x&lt;41x+4if x≥4 To show that f(x) has a jump discontinuity at x = 4.

Q: 1. Use the graph to find the limits In (t + 2) f(t) = 2- -2 2 a) t→0 lim f(t) b) lim f(t) t→-1

A:

Q: Q.2 Show that r(ax) _ 1 lim x0 (x)

A: to show  limx→0ΓaxΓx=1a   ...i as we know Γn=Γn+1n using this formula we get Γx=Γx+1xΓax=Γax+1ax

Q: 3. Consider the function f defined below. Inx for 0<xs2 If f(x)={ then lim f(x) is **]x²In 2 for 2…

A: Given:  fx=lnx         for 0&lt;x≤2x2 ln2     for 2&lt;x≤4limx→2 fx=?

Q: 37. The function / has a removable discontinuity at z„. Therefore A S(7,) exists B lim f(z) exists C…

A:

Q: 1. Show that for all f e L'(R) lim

A: To show: limδ→1∫f(x)-fxδdx=0 Concept used: ∫fxdx=Fx+c Chain rule: ∫fxadx=1aFx+c

Q: 1(a)Find y' given that y In(2x + 1) + e**+5 %3D 5x2-1 b) Evaluate lim x00 2x2+1 c) Find y' if x'+ y…

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Q: -1 sx3 O A) -0 O B) -1 O C) 7 O D) Does not exist

A:

Q: 13. Let f(x) = mx + b, where m and b are constants. If lim f(x) = -2 and lim f(x) 4, find m and b.…

A: Answer 13 has been provided below

Q: Suppose that functions g(t) and h(t) are defined for all values of t and g(0) = h(0) = 0. Can…

A: Given that functions g(t) and h(t) are defined for all values of t and g (0) = h (0) = 0.

Q: 3. If f and are continuous functions with f(3) 4 and lim 2f(x)-9(r)3D5, what is g(3)?

A: To evaluate the value of the function

Q: Let fn → f, where if 0 <x < L3 (3 – x) if 1 < x<1 { fn (x) п-3 if 0 < x < 1 - and f(x) if x = 0 Then…

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Q: Show that the function is continuous at the given value. f(x) = 3x, c = 2 The function f(x) is…

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Q: 2. Let x E (0, 1), use e – d to prove V4 +x – 2 lim 1 4

A: 2. Given x∈0,1, limx→04+x-2x=14                       ... (1) Use ε-δ definition to prove above the…

Q: 1. Show that for all f e L'(R) lim da = 0.

A:

Q: Prove that lim n 00 a" f (x) dx = 0, for any f E R[0, 1].

A:

Q: Find the value of the limit below. [9 - 4(x + Ax)]? – (9 – 4x)2 lim Ax - 0 Ax Specify the function f…

A: We have to find given limit.

Q: h of the following limits analytically; show your work: O LIVE 9-(3-h)2 (a) limp-0 1 1 (b) limx-1…

A: Please refer the attached image for complete solution.

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.