251

500 Rozwiązania i odpowiedzi

21.13. 1

21.15.    /

21.16.    I

21.17.    /

21.18.    I

21.19.    I

21.21. /

= lim [—Vl — *²]o '=1 •

*-► + 0

= lim ln [jc + \/x²— l]f _{+ £} = In (2+_N/3).

E~* + O

= lim [Vx²-4]2_+e=_N/5.

c-* + 0

= lim [3(x²-4)*]²-‘'+ lim [3 (x²-4)*]f_+£2 = 9 »/4.

C| + 0    «2“* + 0

lim [| (x+1) \jx - x² -1 arcsin (1 — 2x)]l~ ^eł = ln.

21.14. /= lm

t-f +o

im,..-

3 3

ci -* + 0 + 0

= lim [-arcsin —1    = |7t.    21.20. /= lim [arcsin—1 .

c-^ + oL    -^Jl+c    €-» + o[_    X J-2

= lim | [arcsin (2x—l)]_e¹I    =

C|-*+0 «2-» +0

21.22. /

21.23.    I

21.24.    /

21.25.    /= lim [i tg 2x]o^/4-'= +oo

c-* +C

21.35. /= lim

lim [| arcsin (2x-l)-\/x-x²]ó"^£=|7t.

E~* +0

lim [-| (x + 5) V(4-x) (x-2)+l arcsin (;r-3)]£^_'=²7i .

e-* + 0

> + o

lim [In (arcsin jr)]|^_,! = ln 3.

21.26. 1= lim [ —ln |cos *|]S^/2_£ = +⁰⁰

E~* + O

f dx    C dx

T—2+ ^1,m    T—-2= ^lim [arctg x]_u + lim [arctg = n

J i i X u-» + co J 1 ~t~ X    u-* — co    y-* + co

u    O

O    v

21.36. / =

f(arctgx)²t/x    f (arctg x)²dx    r(arctgx)^3_|°

l + v» ^{+ hm}    i t y2 - ^lim -r-    +

u — OO J    l i A    v~* + CO J    lT-A    «-*-coj_    _u

—    ¹ *r³

I2ⁿ *

+ lin, u-* + co [_    3

JO

21.37. /= lim [jarctg|A:]“ =I(irc-Ijt)=irt.

y-*+CO    ^{V J}

21.38. ;= lim	ri-^1!	-i.	21.39. /= lim	1 ■
U-* + co	L 3 M J	3	W-» + 00	x-3_

21.40. /= lim i7²[^arctg^\^/,2]“ ⁼ Ś^7CN/²“4\/^2arctgW²-

U~» + co

21.41. 1

= ^iim

li~* + 00 [_    \

arctg -

x-3

= Jtt.

21.42.    /

21.43.    /

21.44.    I

21.45.    I

21.46.    /

21.47.    I

21.48.    / 21.50. I

21.52.    I

21.53.    I

= lim f _v

U-* - 00 [_

/3 arctg

2x+l"|“^ł

-JTl

= 3^b>/3 •

= lim [arctg(* + l)]°+ lim [aictg(x-l)]o = Jt-

M-* — CO    V~* + OO

= Hm T-L+ln—l =-i

,[_x + l    x + lj, ²

lim

u-* + oo I

= lim

V — + oo

r,, U+*r i 2*-

Ig ^ln J3T7T2⁺-^^arct8

-x+x² ^/3

V3

[1-7TI2 ■ 75—‘*2V3Jo

= lim T — arcsin—1    + lim —

e-»0 [_    -XJl+c u-> + 00 |_

ln ^5>.- +4= arctg-^-=] =±nj3-'₂\n£.

= lim — e‘“] =—.    21.49. /= lim [-i<r*²]S = i.

v-» + «[_    ^ Jo &    u-*+co

= lim [ — -le^_'^c(cosx + sinx]o = j.    21.51. /= lim [ —=e -1.

V~* + 00    M “* + 00

i

sfa^b-

-arctg

/ ia-b

W

”[Ś ^arC'⁸ (t ^,E*)]o -

K

lab

-arcsin

Jo Vn²-i>²