Решение
1) lim
x-->0 [(√(7 - x) - √(7 + x))*(√(7 - x) + √(7 + x))] / (√7 *
x) =
= lim x-->0 [(√(7 - x)² - (√(7 + x))²] / (√7 * x) =
lim
x-->0 (7 - x - 7 - x) /
(√7 * x) = lim x-->0 (- 2x) / (√7 * x) =
= -
2 / √7 = - 2√7 / 7
2)
lim x-->-1 [(√5+ x) - 2)*(√(5 + x) + 2)*√(8 - x) + 3)] /
/
[(√(8 - x) - 3)*(√(8 - x) + 3)*(√(5 + x) + 2)] = lim x-->-1 [(5 + x - 4)*
* √(8
- x) + 3)] / [(8 - x - 9) * (√(5 + x) + 2)]
= lim x-->-1 [(1 + x)*(√(8 - x) + 3)] / [(11 -
x)*(√(5 + x) + 2)] = 0
3) lim x-->4 [(2 - √x)*(2 + √x)*(√(6x + 1) + 5) ] / [(6x
+ 1) - 5)*
* (√(6x + 1) + 5)* (2 + √x)] = lim x-->4 [(4 - x)*(√(6x + 1) +
5) ] /
/ [(6x + 1 - 25)* (2 + √x)] = lim x-->4 [(4 - x)*(√(6x + 1) +
5)] /
/
[6*(x - 4)* (2 + √x) = - lim x-->4 [(4 - x)*(√(6x + 1) + 5)] /
[6*(4 - x)* (2 + √x)] = - lim x-->4 [(√(6x + 1) + 5)] / [6* (2 + √x)] = 10/24 =
= 5 /12
4) lim x--> -1 [√(x + 20) - 4)*(√(x + 20) + 4)] / (x³+ 64) =
lim x--> - 1 (x + 20 - 16) / (x³+ 64) = lim x--> - 1 (x + 4) / [(x+ 4)*(x² - 4x + 16)] = lim x--> -1 [1 / (x² - 4x + 16)] = 1/21
5) lim x--> 3 (2x³- 3x - 9) / [(√x - 2) - √(4 - x)] =
2x² - 3x - 9 = 0
D = 9 + 4*2*9 = 81
x = (3 - 9)/4 = - 3/2
x = (3 + 9)/4 = 3
lim x--> 3 [2*(x + 1,5)*(x - 3)*(√(x - 2) + √(4 - x)] /
/ [(√(x - 2) - √(4 - x)) * (√(x - 2) + √(4 - x))] =
lim x--> 3 [2*(x + 1,5)*(x - 3)*(√(x - 2) + √(4 - x))] / (x - 2 - 4 + x) =
= lim x--> 3 [2*(x + 1,5)*(x - 3)*(√(x - 2) + √(4 - x))] / [2*(x - 3)] =
= lim x--> 3 [2*(x + 1,5)*(√(x - 2) + √(4 - x))] / 2 =
= (2*4,5*2) / 2 = 9