Use series to evaluate the limits in Exercises 29-40. e* – (1 + x) e – e* 29. lim 30. lim x→0 x2 x→0 х cos t (F/2) sin 0 – 0 + (0³/6) 31. lim 32. lim 1→0 0→0 y tan-ly tan-ly – sin y 33. lim y→0 34. lim y→0 y° cos y 35. lim x? (e-1/x² – 1) 1 36. lim (x + 1) sin x + 1 In (1 + x²) x – 4 37. lim x→0 1 – cos x 38. lim x→2 In (x – 1) sin 3x2 In (1 + x³) 39. lim x→0 1 - cos 2x 40. lim x→0 x•sin x²

Q: Let f(x) be periodic with period 2pi and defined on [0, 2pi] as follows:f(x) = 1 if 0 ≤ x ≤ pi/2f(x)…

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Q: Find Lim (x² (X, Y) → (0, 0) + Y²) Log (x² + Y²)

A: To find the limit  lim(x,y)→(0,0)(x2+y2)log(x2+y2)

Q: Exercise 2: Let f(x) = sec2(x), and g(x) = tan2(x). Prove f'(x) = g' (x). Thus, find a C E R such…

A: Given,

Q: lim x2 Vx2 e) lim xe? In2 x-5 ln a+6 In x-2 Ps

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Q: If f(2010) = π/2, f(2x) = f(x)Vx E R and f(x) is continuous for x E R, then the value of…

A: Let us solve above problem by using L'Hopital rule

Q: + Vx + vx Q3: a) Differentiate b) Differentiate (Cosx)³tnx with respect to (Tanx)Secx ´with respect…

A: a)  f(x) = x+x+x      f'(x) = 12x+x+xd(x +x+x)dx             = 12x+x+x × (1+12x +x)×d(x+x)dx…

Q: Sin@t) Sin (2+) In (tan (2€)) In Con (8t)) Let F(t)= (t + 1) Cos (3€) Find the limit F(t) tso

A: Given,           Ft=sin3tsin2ti+lntan2tlnsin3tj+t+1cos3tk

Q: lim, ol cos(a)x – cos (b)x

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Q: Sinx e* lim x→0 x- Sinx (b) If x = cos0,y = cosm0;m is an integer greater than unity, prove that %3!…

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Q: o lim x-0 x² - sin²2x 2 x 2

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Q: x² + 9 dx J x2 – 1)(x² + 4) | In \x + 1| + In |x – 1| +; tan-1 (-) + c -In |x + 1| - In |x – 1| -;…

A: Topic:- application of integration

Q: (1) Use the definition to prove that if xn → 2 then (2xn + 1) / xn → 5/2.

A: As per our company guideline we are supposed to answer only first three  qs, kindly post remaining…

Q: -tan (m + 1)(x) + C tan I + m um + ( x ) (x + ₁)² I m+3 1 с

A: Using substitution then integrating we get correct answer as (C)

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…

Q: Find the discontinuities, if any.  f(x) = sin (x^2 - 2)  f(x) = sec x lim    sin θ / θ2 θ→0+

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Q: sin-1(e* 2 lim V1 - e2x x→0-

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Q: P2: |(e* sin x + x In x) dx

A:

Q: e* – 1 lim x-0 tan 2x

A: Given: limx→0ex-1tan2x

Q: Evaluate: / [sec (x) – ese²(x) ]dv А tan (x) - сo(х) +с ® ;o*(x) + w«*(+) + Cc (B sec (x) – csc (x)…

A: We have to solve this integration