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в) lim ( + 5 ) ; г)lim( )
x→+∞ x→

0
a) lim ( ) = 0 = ;
x→+0

ln x ∞ (ln x ) '
ln A= lim ln ( ) = lim = = lim =
3x
x→+0 x→+0 ln (e -1) ∞ x→+0 ( ln (e -1)) '
3x

1 3x
1 e -1 0
3x
= lim = lim : lim e = =
x→+0 1 ∙ e 3x 3 x→+0 x x→+0 0
3x
e -1

3x
3x
1 (e − 1) 1 e ·3
1
= lim : 1 = lim =1; A=e =e;
3 x→+0 3 x→+0 1
4 4 ln(cos 3x) 0
∞
2
2
б) lim(cos 3x)x =|1 |=A; ln A = lim ln(cos 3x)x =4 lim = =
x→0 x→0 x→0 x 2 0

1
ln(cos 3x) ' cos3x ·(-sin3x)·3 tg3x 0
= 4 lim =4 lim =-6 lim = =
x→0 2 ' x→0 2x x→0 x 0
(x )

1
( 3 ) cos3x ·3
= -6 lim = -6 lim = −18; = ;
x→0 x→0 1

x
1 1 ln(x+5 ) ∞
x 0 x
в) lim (x+5 )x = ∞ =A; ln A= lim ln(x+5 )x= lim = =
x→+∞ x→+∞ x→+∞ x ∞

1 x
x ' x ·(1+5 ln 5) x
(ln (x+5 )) x+5 1+5 ln 5 ∞
= lim = lim = lim x = =
x→+∞ x ' x→+∞ 1 x→+∞ x+5 ∞

x ' x x '
(1+5 ln 5) ln 5 ·5 ln 5 ∞ (5 )

= lim = lim x = = ln 5· lim x =
x '
x→+∞ (x+5 ) x→+∞ 1+5 ln 5 ∞ x→+∞(1+5 ln 5) '
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