2. 2 I,II,III) 2 x expx) = lim + x 3) ) expx) e x 3) x. ) {a } a a 2 a 3...) a b b {a } α : lim a = α b) ) [] 2 ) f x) = + x ) 4) x > 0 {f x)} x > 0,

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1 a x x ) a x x a x x a x x ) a x x [] 3 3 sup) if) [3] 3 [4] 5.4 ) e x e x = lim + x ) ) e x e x log x = log e x) a > 0) x a x = e x log a 2)

2 2. 2 I,II,III) 2 x expx) = lim + x 3) ) expx) e x 3) x. ) {a } a a 2 a 3...) a b b {a } α : lim a = α b) ) [] 2 ) f x) = + x ) 4) x > 0 {f x)} x > 0, f x) = + ) x < f+ x) = + x ) + 5) +

3 2. 3 f x) f + x) + ) k 0 k ) 2 f x) x k b k b k = C k k 6) k 2 b ) 2) k + ) k = k! k = k! 2 k + = ) 2 ) k ) k! 7) f + x) x k b + k b + k = ) 2 ) k ) k! k b k < b + k k = 0, b 0 =, b = C = b k = b + k f + x) + ) x + + ) + = 5) x > 0 f x) b k b k k! 2 k 0 k ) 8) k = 0, b 0 = b =, 0! = k 2 7) b k /k! k 2 /k! /2 k k! = 2 3 k = 2 k

4 2. 4 8) x > 0 2x < N N N f x) A A A N, x N, x N {f x)} A, f x), f 2 x),... f N x) B B f x) B x {f x)} N 8) f x) = + x ) = b kx k k=0 x < N/2 k N k x k k! = xn N )! f x) N k=0 k=n 2 k=0 x k N+ NN + ) k x k k! + k=n x k N k! k=0 x k k! xn N )! xk N+ N k N+ x k k! + xn N )! k=n 2 ) xn k N+ N )! 2 ) k N+ ) k N+ = N+ = 2 N+ f x) f x) N k=0 x k k! + xn N )! x > 0 3)

5 2. 5 x = 0 3) x < 0 x = y > 0) f x) = + x ) = y ) ) y = = ) = y + y ) = + y y ) + y y + y = + y ) 9) y + y ) ) 2 0) 2 y 2 c y) = { 2 / 2 y 2 )} 2. ). {a }, {b } a b lim a = α, lim b = β α, β ) α β 2. {a }, {b }, {c } a b c lim a = lim c = α lim b α 2 [] ) > 2 y 2 > 2y 2 2 y 2 > = 2 2, 2 y 2 < 2 c y) = { 2 / 2 y 2 )} = { + y 2 / 2 y 2 )} > 2y < c y) < + 2y2 2 ) ) 3 2 c y) 3. a > 0 lim a = α > 0 lim a = 2)

6 a = a / a x x = 0 3 3) lim a = α lim a α = 0 α > 0 a α < α/2 [] ) α/2 < a α < α/2 α 2 < a < 3α 2 3) 3) α 2 < 3α a < 2 4) β > 0) lim β = 5) 4) 2) 5) β = β = 5) β > β > = β = p p > 0 β = + p ) = + p + + p + p β p > 0 lim p = 0 5) 0 < β < /β = γ γ > β = γ = γ

7 2. 7 /γ ) 0 < β < 5) 3 ) ) + 2y2 2 ) = + 2y2 2 ) 2, lim + 2y2 2 ) 2 = exp2y 2 ) y 2 > 0 3 lim + 2y2 2 ) = lim c y) = 0) 0) expy) 9) lim f x) = expy) = exp x) x < 0 3) expx) = exp x) 6) e x = /e x 6) x c y) 4. A, B ) B lim + = 7) 2 + A 2 A 2 + A 2 A = 2 2

8 3. 8 B A + B 2 + A + 2 B, 2 B A B 2 + A 2 B 2 > 2 B 2 ) + 2 B 2 B / 2 > 0 ) B + 2 B ) 2 + A 2 lim 2 B 2 ) 2 = exp 2 B ), lim + 2 B 2 ) 2 = exp2 B ) 3 7) 3 expx) x < y + x ) < + y ) expx) expy) x < y) 8), x > 0 + x ) = b kx k + x k=0 expx) + x x > 0) 9)

9 3. 9 x < 0 6), 9) expx) = exp x) x 20) 6), 9) x expx) 9), 20) lim expx) =, lim x expx) = 0 2) x ) m { + ) x } m = + x ) m = + mx m ) m expx) m = expmx) 22) 6) expx) = m expmx) = exp mx) m expmx) = expx) m expx) m m > 0 ) m < 0 ) 23) k 22) expx) = exp k x ) ) x k = exp k k ) x exp = k expx) k

10 ) m, k k > 0) ) m exp k x = k expx) m k expx) m m > 0 ) m < 0 ) 24) q e x ) q = e qx y e x ) y = e xy a a x expx) expy) + x ) + y = ) = = + x + y + x + y + x + y + xy ) 2 ) + x + y + x + y ) + + xy 2 xy 2 + x + y) ) 4 expx + y) expx) expy) = expx + y) 25) e x e y = e x+y x > 0 9) expx) > 25) x < y expy) = expy x) expx) > expx) 26) 8) 25) y y 6) expx) expy) = expx y) 27) e x /e y = e x y

11 4. 4 a > 0) a x powa, x) a = expp) p powa, x) = exppx) 28) a p 3 26) expx) 2) a > 0 a = expp) p p a a p a = expp) p = exp a) ) p = la) 28) powa, x) = expx la)) 29) 2) 28) a > 0, x a x powa, x) p = e = exp) = le) powe, x) = exp x) = expx) expx) e = exp) x powa, x) 28) p > 0 powa, x) p < 0 powa, x) exp0) = p = la) > 0 a > p < 0 0 < a < 2) a > powa, x) powa, x) =, lim powa, x) = 0 lim x x 0 < a < powa, x) powa, x) = 0, lim powa, x) = lim x x powa, 0) = exp0) =, powa, ) = expp) = a 30)

12 4. 2 x x = m/ m, ) 24) pow a, m ) ) m = exp p = expp) m = a m x x = m/ pow a, m ) = exp m ) p = expp) = m a m powa, x) a x 25), 27) powa, x) powa, y) = exppx) exppy) = exppx + y)) = powa, x + y), powa, x) = exppx) = exppx y)) = powa, x y) powa, y) exppy) a x a y = a x+y, a x /a y = a x y ) powa, x) powb, x) p = la), q = lb) a = expp), b = expq) ab = expp) expq) = expp + q) p + q = lab) powab, x) = expp + q)x) = exppx + qx) = exppx) expqx) = powa, x) powb, x) ab) x = a x b x ) p q = la/b) ) a pow b, x = powa, x) powb, x) a/b) x = a x /b x ) a x ) y powpowa, x), y) p = la), q = lpowa, x)) a = expp), powa, x) = expq) powa, x) = exppx) = expq)

13 5. 3 expx) q = px powpowa, x), y) = expqy) = exppxy) = powa, xy) a x ) y = a xy 5 expx) x x + x ) x x 2 3) + x ) ) x k = C k = + x + b k=0 k x) k k=2 x 8) b k x) k k=2 b k x k k=2 x 2 k=2 x 2 = x 2 2 k ) 2 3) exp x) x x 2 x ) 32) expx + x) expx) x = expx) exp x) expx) x = expx) exp x) x 32) expx + x) expx) x expx) = expx) exp x) x expx) x x x 0 0 expx + x) expx) lim x 0 x = expx)

14 6. 4 expx) expx) {expx)} = expx) 33) dy dx = dx dy y = expx) x = ly) d ly) dy = dy dx = expx)) = expx) = y l) = 0 expx) ly) ) ly) = y dt t 6 expx) 3) 3) x z = x + y i x = Re z, y = Im z ) α = p + qi lim x = p lim y = q z = a + bi 0) r = z = a 2 + b 2 > 0) θ = argz) ) a = r cos θ, b = r si θ z = a + bi = rcos θ + i si θ) z

15 6. 5 cos α + i si α)cos β + i si β) = cos α cos β si α si β + icos α si β + si α cos β) = cosα + β) + i siα + β) 34) cos θ + i si θ) = cos θ + i si θ) ) 35) 3) x x = iα α ) + iα/ + iα = + α2 2 cos β + i si β ), β = arg + iα ) β π/2 < β < π/2 r cos β =, r si β = α/ ta β = α/ f iα) = = + iα ) = + α2 cos β 2 + i si β ) ) + α2 cos β 2 + i si β ) 36) 4 ta β = α/ 0 β 0 ta x lim x 0 x = β = ta β β β = α α ) ta β ta β f iα) lim f iα) = lim + iα ) = cos α + i si α 37)

16 6. 6 expx) 3) expiα) = cos α + i si α 38) e iα = cos α + i si α x z = x + iy x, y ) + x + iy = + x ) 2 y 2 + cos γ 2 + i si γ ), γ = arg + x + yi ) > x + x/ > 0 π 2 < γ < π 2, ta γ = y + x = y + x 0 ) lim γ = 0 γ = ta γ γ ta γ = y + x γ ta γ y ) { + x ) 2 y 2 } + = + 2x 2 + x2 + y 2 ) 2 = + 2x ) + 2x + x2 + y x = = + 2x ) + x2 + y 2 ) 2 + 2x + 2x ) + x 2 + y x 4 exp2x) lim + x + iy ) = lim { + x ) } 2 y 2 + cos γ 2 + i si γ ) = exp2x) cos y + i si y) = expx)cos y + i si y) 24) expz) = expx + iy) = expx)cos y + i si y) 39)

17 7. 7 e 39) expz) expw) = expz + w), expz) expw) = expz w), expz) = expz) 40) ) 7 III e x e x 2) 29)) e x 29) 3) 34) 3) [] 325) 98) [2] 999)

18 7. 8 [3] E. ) e 999) [4] ) 996) [5] E. G. ) 202)

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z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy f f x, y, u, v, r, θ r > = x + iy, f = u + iv C γ D f f D f f, Rm,. = x + iy = re iθ = r cos θ + i sin θ = x iy = re iθ = r cos θ i sin θ x = + = Re, y = = Im i r = = = x + y θ = arg = arctan y x e i =

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1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ = 1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A

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