lim f(x) lim g(x) 0, lim f(x) g(x),

2016 11 14 1 15

lim f(x) lim g(x) 0, lim f(x) g(x), 0 0. 2 15

1 f(x) g(x) (1). lim x a f(x) = lim x a g(x) = 0; (2). a g (x) f (x) (3). lim ( ). x a g (x) f(x) lim x a g(x) = lim f (x) x a g (x). 3 15

.. 4 15

0 0 0 1 0 0 0. 5 15

n N +, lim x n ln x x 0 + 6 15

n N +, lim x n ln x ln x = lim x 0 + x 0 + x n 6 15

n N +, lim x n ln x ln x = lim x 0 + x 0 + x n L == lim x 0 + 1/x nx n 1 6 15

n N +, lim x n ln x ln x = lim x 0 + x 0 + x n L == lim x 0 + 1/x nx n 1 x n = lim x 0 + n = 0. 6 15

( 1 lim x 0 x 1 ) sin x 7 15

( 1 lim x 0 x 1 ) sin x = lim x 0 sin x x x sin x 7 15

( 1 lim x 0 x 1 ) sin x sin x x = lim x 0 x sin x sin x x = lim x 0 x 2 7 15

( 1 lim x 0 x 1 ) sin x sin x x = lim x 0 x sin x sin x x = lim x 0 x 2 L == lim x 0 cos x 1 2x 7 15

( 1 lim x 0 x 1 ) sin x sin x x = lim x 0 x sin x sin x x = lim x 0 x 2 L cos x 1 == lim x 0 2x x 2 /2 = lim x 0 2x 7 15

( 1 lim x 0 x 1 ) sin x sin x x = lim x 0 x sin x sin x x = lim x 0 x 2 L cos x 1 == lim x 0 2x x 2 /2 = lim x 0 2x = 0 7 15

( 1 lim x 0 x 1 ) = x 2 + x 8 15

( 1 lim x 0 x 1 ) x 2 + x x = lim x 2 + x x 0 x(x 2 + x) 8 15

( 1 lim x 0 x 1 ) x 2 + x x = lim x 2 + x x 0 x(x 2 + x) x 2 = lim x 0 x(x 2 + x) 8 15

( 1 lim x 0 x 1 ) x 2 + x x = lim x 2 + x x 0 x(x 2 + x) x 2 = lim x 0 x(x 2 + x) 1 = lim x 0 x + 1 8 15

( 1 lim x 0 x 1 ) x 2 + x x = lim x 2 + x x 0 x(x 2 + x) x 2 = lim x 0 x(x 2 + x) 1 = lim x 0 x + 1 = 1. 8 15

lim x sin x x 0 + 9 15

lim x sin x (sin x) ln x = lim e x 0 + x 0 + 9 15

lim x sin x (sin x) ln x = lim e x 0 + x 0 + = e lim x 0 + (sin x) ln x 9 15

lim x sin x (sin x) ln x = lim e x 0 + x 0 + = e lim x 0 + (sin x) ln x = e lim x 0 + x ln x 9 15

lim x sin x (sin x) ln x = lim e x 0 + x 0 + = e lim x 0 + (sin x) ln x = e lim x 0 + x ln x = e 0 = 1. 9 15

3.4 10 15

y = f(x) [a, b] (a, b).. 11 15

y = f(x) [a, b] (a, b).. (1). (a, b) f (x) 0 f(x) [a, b]. 11 15

y = f(x) [a, b] (a, b).. (1). (a, b) f (x) 0 f(x) [a, b]. (2). (a, b) f (x) 0 f(x) [a, b]. 11 15

y = f(x) [a, b] (a, b).. (1). (a, b) f (x) 0 f(x) [a, b]. (2). (a, b) f (x) 0 f(x) [a, b]. ( ). 11 15

[a, b] x 1 < x 2. 12 15

[a, b] x 1 < x 2. x (x 1, x 2 ), f (x) > 0, 12 15

[a, b] x 1 < x 2. x (x 1, x 2 ), f (x) > 0, ξ (x 1, x 2 ) f(x 2 ) f(x 1 ) = f (ξ)(x 2 x 1 ) > 0, 12 15

[a, b] x 1 < x 2. x (x 1, x 2 ), f (x) > 0, ξ (x 1, x 2 ) f(x 2 ) f(x 1 ) = f (ξ)(x 2 x 1 ) > 0, f(x 2 ) > f(x 1 ). 12 15

[a, b] x 1 < x 2. x (x 1, x 2 ), f (x) > 0, ξ (x 1, x 2 ) f(x 2 ) f(x 1 ) = f (ξ)(x 2 x 1 ) > 0, f(x 2 ) > f(x 1 ). x (x 1, x 2 ) f (x) = 0, 12 15

[a, b] x 1 < x 2. x (x 1, x 2 ), f (x) > 0, ξ (x 1, x 2 ) f(x 2 ) f(x 1 ) = f (ξ)(x 2 x 1 ) > 0, f(x 2 ) > f(x 1 ). x (x 1, x 2 ) f (x) = 0, z 1 < z 2, < z n (x 1, x 2 ) 12 15

[a, b] x 1 < x 2. x (x 1, x 2 ), f (x) > 0, ξ (x 1, x 2 ) f(x 2 ) f(x 1 ) = f (ξ)(x 2 x 1 ) > 0, f(x 2 ) > f(x 1 ). x (x 1, x 2 ) f (x) = 0, z 1 < z 2, < z n (x 1, x 2 ) f(x 2 ) > f(z n ) > f(z n 1 ) > > f(z 1 ) > f(x 1 ), 12 15

[a, b] x 1 < x 2. x (x 1, x 2 ), f (x) > 0, ξ (x 1, x 2 ) f(x 2 ) f(x 1 ) = f (ξ)(x 2 x 1 ) > 0, f(x 2 ) > f(x 1 ). x (x 1, x 2 ) f (x) = 0, z 1 < z 2, < z n (x 1, x 2 ) f(x 2 ) > f(z n ) > f(z n 1 ) > > f(z 1 ) > f(x 1 ), x 1 < x 2 f(x 2 ) > f(x 1 ). f(x) [a, b]. 12 15

y = x sin x. 13 15

y = x sin x. y = 1 cos x 0 13 15

y = x sin x. y = 1 cos x 0 x R 13 15

y = x sin x. y = 1 cos x 0 x R [a, b], y [a, b]. 13 15

y = x sin x. y = 1 cos x 0 x R [a, b], y [a, b]. y [a, b]. 13 15

y = x sin x. y = 1 cos x 0 x R [a, b], y [a, b]. y [a, b]. y = x sin x (, + ). 13 15

y = 2x 3 6x 2 18x 7. 14 15

y = 2x 3 6x 2 18x 7. y = 6x 2 12x 18 = 6(x 3)(x + 1), 14 15

y = 2x 3 6x 2 18x 7. y = 6x 2 12x 18 = 6(x 3)(x + 1), x (, 1) (3, + ) y > 0; 14 15

y = 2x 3 6x 2 18x 7. y = 6x 2 12x 18 = 6(x 3)(x + 1), x (, 1) (3, + ) y > 0; x ( 1, 3) y < 0. 14 15

y = 2x 3 6x 2 18x 7. y = 6x 2 12x 18 = 6(x 3)(x + 1), x (, 1) (3, + ) y > 0; x ( 1, 3) y < 0. (, 1] [3, + ), [ 1, 3]. 14 15

y = 2x 3 6x 2 18x 7. y = 6x 2 12x 18 = 6(x 3)(x + 1), x (, 1) (3, + ) y > 0; x ( 1, 3) y < 0. (, 1] [3, + ), [ 1, 3]. (, 1] [3, + ). 14 15

x ( 0, π ) sin x + tan x > 2x. 2 15 15

x ( 0, π ) sin x + tan x > 2x. 2 f(x) = sin x + tan x 2x, 15 15

x ( 0, π ) sin x + tan x > 2x. 2 f(x) = sin x + tan x 2x, f(0) = 0, x ( 0, π ) 2 f (x) = cos x + sec 2 x 2 15 15

x ( 0, π ) sin x + tan x > 2x. 2 f(x) = sin x + tan x 2x, f(0) = 0, x f (x) = cos x + sec 2 x 2 = cos x + 1 cos 2 x 2 ( 0, π ) 2 15 15

x ( 0, π ) sin x + tan x > 2x. 2 f(x) = sin x + tan x 2x, f(0) = 0, x f (x) = cos x + sec 2 x 2 = cos x + 1 cos 2 x 2 1 2 cos x cos 2 x 2, ( ) ( 0, π ) 2 15 15

x ( 0, π ) sin x + tan x > 2x. 2 f(x) = sin x + tan x 2x, f(0) = 0, x f (x) = cos x + sec 2 x 2 = cos x + 1 cos 2 x 2 1 2 cos x cos 2 x 2, ( ) 1 = 2 2 = 2 1 cos x > 0. cos x cos x ( 0, π ) 2 15 15

x ( 0, π ) sin x + tan x > 2x. 2 f(x) = sin x + tan x 2x, f(0) = 0, x f (x) = cos x + sec 2 x 2 = cos x + 1 cos 2 x 2 1 2 cos x cos 2 x 2, ( ) 1 = 2 2 = 2 1 cos x > 0. cos x cos x f(x) [0, π/2]. ( 0, π ) 2 15 15

x ( 0, π ) sin x + tan x > 2x. 2 f(x) = sin x + tan x 2x, f(0) = 0, x f (x) = cos x + sec 2 x 2 = cos x + 1 cos 2 x 2 1 2 cos x cos 2 x 2, ( ) 1 = 2 2 = 2 1 cos x > 0. cos x cos x f(x) [0, π/2]. f(0) = 0, ( 0, π ) 2 15 15

x ( 0, π ) sin x + tan x > 2x. 2 f(x) = sin x + tan x 2x, f(0) = 0, x f (x) = cos x + sec 2 x 2 = cos x + 1 cos 2 x 2 1 2 cos x cos 2 x 2, ( ) 1 = 2 2 = 2 1 cos x > 0. cos x cos x ( 0, π ) 2 f(x) [0, π/2]. f(0) = 0, f(x) > 0 (0, π/2). 15 15

x ( 0, π ) sin x + tan x > 2x. 2 f(x) = sin x + tan x 2x, f(0) = 0, x f (x) = cos x + sec 2 x 2 = cos x + 1 cos 2 x 2 1 2 cos x cos 2 x 2, ( ) 1 = 2 2 = 2 1 cos x > 0. cos x cos x ( 0, π ) 2 f(x) [0, π/2]. f(0) = 0, f(x) > 0 ( (0, π/2). x 0, π ) sin x + tan x > 2x. 2 15 15