49. Prove that x +> 2 (x1/2 – x-1/2)2. 2 for all x > 0. Hint: Consider

Q: B. Prove the fillang tim Jn2 +n 4- (6) lim n-700 Lvntl trn

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Q: 9. For |x|<1, prove that 8x 16x 24x 1-x1+x 1-x 2 8x 8x = 1+ (1 – x)? (1+x2 (1-x)? ...

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Q: Prove that 2 1 1 5 2 (4-2x)² for all relevant x-values

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Q: 7. There is aremovable discontinuity at x = 1 andf(1) = 4

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A: The given functIon is: f(x)=14x2-2x

Q: 9. If f(x) # 0f and f(x) is relatively prime to Or, what can be said about f(x)?

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A: We will find the limit.

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A: Let's Solve,

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Q: 4. Suppose that f(2) = 16 and that f'(x) 2 – 3 for all x in the interval [2,7]. Determine the…

A: 4.f(2)=16 and that f'(x)≥3 for all x in the interval 2,7

Q: if ニ if 1-X fCx +2) = f(x) Hx f on [-3,3] and we Suppose draw the funch'on

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Q: 8. )Prove that lim Vr = 2 using the e-8-definition. Hint: VI – 2 = x- 4 VI + 2 Puouo thot thoro…

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Q: 8. What is the value of c guaranteed to exist by the Mean Value Theorem if m(x) = x4 - 3x over the…

A: The answer for the above question is as shown below.

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Q: Prove or provide a counterexample to the following if r ∈ ( 0 , 1 ), then lim n→∞ r^n exists.

A: limn→∞an=L if for every number ε&gt;0 there is an integer N such that                   |an-L|&lt;ε…

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Q: Prove the following inequality. Do not evaluate the integral. 1 ,1 - x² + x +1 | - 2 0, 1+ x4 4

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Q: Excrcise 8. Evaluate -In 2 xe d.x 1

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Q: Q 4. (a). Show that limit *1, does not exist if (5-4x S(x)=. -)- {ax-2 for x <1] for xz1]

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Q: 5. Show that Ax) = Inx² - 1- In/3, if f'(x)= and (2) = 0. x- 1 Justify your answer.

A: To show that f(x)=ln|x²-1|-ln√3

Q: (х — т 11- тх if x < 3 if x 2 3 5) Find m so that g(x) = Is continuous for all x?

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Q: Prove that (0.01)(100sin2π(5000)t)2 is equal to 0.5+0.5cos2π(10000)t

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Q: 6. Prove from first principles (with ac- 6 argument ) that लेग ½ lim 7--1

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Q: 21. Discuss the continuity of : 1 x + for x 2

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Q: a) Verify Rolle's Theorem for ƒ(x)= 2.x' – 8.x + 4 over the interval [0,2].

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Q: 53. Show, for f(x) = 3x2 + 4x over [0, 2], that ( 12j², 8j Σ%) RN N2 Then evaluate lim RN.

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Q: [6] 1. Use the definition of the definite integral as the limit of a sum to evaluate (4r – r) dr.…

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Q: any n > N(\epsilon) | fn(x) - f(x) | < \epsilon

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Q: 5) Find m so that g(x) (x - m =1-mx if x <3 if x 2 3 Is continuous for all x?

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Q: 29. For |x|<1, prove that 16x 24x 8x 1 + 1-x1+x 1-x ..... .. 8x 8x = 1+ (1- x)? (1+x)²' (1-x)?

A: