x + 3 6x2 + x - 1 41. lim x-3 x +x – 6 42. lim x→(-1/2)* 4x² - 4x - 3 43. lim 44. lim 6 45. lim x→-4 x² + x + 4, 46. lim r→3*\3 TTX + cot -2 47. lim x0+ Sin x lim 48. x→(7/2)* COs x e 50. lim In(x2 – 16) lim x8- (x-8)3 - 51 lim In cos x| 52. lim e-0.5.t sin x x(7/2) 75% 31°C Hujan

Q: funetion:- sinty -X 1-COS X tank t cos X メ-secK X zo Dcontinous at Xz0 and equals Dcontinous at X=0…

A: Given  f(x) = sin2x - x1 - cos x,x<0tan x + cos xx - sec x,x≥0 We have to check the continuity of…

Q: 25) Use Dr. Dai's Differential Table to find [tan - 1(x) ]' =, 25) A) 1/ (1+ x2) B) – tan-2(x) sec2x…

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Q: Differentiation#11

A: We have

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A:

Q: 16. f(x) = x' + x + 4 on (-1,2). 17. f(x) = x -x² – 30x + 3 on (0, 6). 18. f(x) = 3 sinx on [T/4,…

A:

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A:

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Q: #35

A: 35) to differentiate the given function

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A:

Q: Increasing and decreasing functions Find the intervals onwhich f is increasing and decreasing.

A: We'll answer the first question only since the exact one wasn't specified. Please submit a new…

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A:

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A: We have to choose the correct option according to the given information

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A:

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A:

Q: Differentiate y=excos(x2): y'= A. excos(x2)-2xsin(x2) B. -2xexsin(x2) C. ex(cos(x2)-2xsin(x2))…

A:

Q: 2 сos(6x) — 6 2? — 2 os(6x) – 6 x² sin(5x) – x2 – 5x - | lim x → 0 - -

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A:

Q: |- lim メ→ X-3 2. lim メ→2 (会 4 メー2 メ*-4 3. lim メ→。 3メ - S;u4X 4. lim メ→。 tau3X 5- lim 1- Cose Sin'e

A: Since you have asked multiple questions. So, we are solving first few for you. If you want any…

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Q: x² +9 √(x² - 1)(x^² + 4) In 1x + 11 + In |x-1| + tan-¹) + C Option 1 In x +11+ In |x-1|-tan ¹ () + C…

A:

Q: The limit in lim (46 x; - 28 sin (x;)) Ax i=1 on the interval o, is equal to 2 (23 x² + 28 cos (x))…

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Q: 1-2x²=1+x+x²³ 2²-2²-1+2+ -2(-1) (2n + 1)! x²+1 sin x- ---- cos x=(-1). - (2n)! −2+4- 6! tan 'x-…

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Q: 2y-n у-п lim п cos y y-

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A:

Q: x3 + x2 ) lim x→-1 x+ 1| (i) ( 3 Vx2 +x +6+ Vx + 8 - 5x (ii) 6) lim x1+ x2 – x (iii) ( sin(3x)…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

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Q: Use Variation of Parameters to solve the differential equation. y"y tanx y C1cos xC2sin x - In secX+…

A: Result used:

Q: 1) f(x) = x5-x¹ + x² - x +² 2) f(x) = sin³x - e³x² + ln(1 + 5x²) 3) f(x)=√1-x² sin-¹ x 4) f(x)=√√x +…

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Q: 1) / sen(x) 4+ cos²(x) I 1+4x² 2) [√5/2 S² 3) 0 4) 1 x² - 2x + 2 dx √ √2/2 are 0 X dr √2/2 arccos(x)…

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Q: 2 sin (4x) lim 2. x2 (1+ cos x)

A: Since you have asked multiple questions in single request  so we  will be answering only first…

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Q: 1.) lim 2x sin'x. *P0 ex - x - 1

A: We apply l- hospital rules and solve it. Follow 2nd step.

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A:

Q: Solve the exact DE (2xy³ + cosx tan y +2 sin x)dx + (3x²y² +sin x sec² y-3 csc y cot y)dy = 0 (1)…

A:

Q: 6x² + 5x + 16 S x3 + 3x² + 2x + 6 5 Injx+31+ In (x² + 2)√2 tan ¹() + C Option 1 5 Injx +31+ In (x² +…

A:

Q: 2x² + x + 3 x³ + x dx In[x] + In (x² + 1) + tan^³(x) + C Option 1 3 In]x] +In (x² + 1) + tan¹(x) + C…

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Q: cos x 13. lim x-(7/2)* 1 – sin x e21 – 1 15. lim - -0 sin t 1 - sin 0 17. lim 0/2 1 + cos 20

A:

Q: 5.f(x) = x (x - 1)(*- 2)(* + 1) As x + +o, f(x) = As x - -n, /(x) As x > I co, f(x) – As x > w./(x)…

A: Solution is given below:

Q: 2x +5x-3 1. lim 6x -7x + 2 4x-6x+3 lim 16x +8x-7 4. x'+8 7. lim 2x-16 1 -1) h-0 hV1+h 10. lim ) sin…

A: 1) Here, the direct substitution  of the limit value will lead the denominator of the function to be…

Q: 1-cos 2X lim x→0 x2 sec x-tan x lim X→- X- 2 x2+sin 3x lim r20 2x+tan 2x

A:

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.