Page 152 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 1
4. (i) Let f(x) = , x  0. Show that lim f(x) = +, lim f(x) =  and lim f(x) = + .
x x + x 0– x 0
-
0
1
(ii) Let f(x) = – , x  0. Show that lim f(x) = –, lim f(x) = – and lim f(x) = –.
x x + x 0– x 0
0
-
1
(iii) Let f(x) = , x  0. Prove that lim f(x) = +, lim f(x) = –.
x x + x 0–
0
1
(iv) Let f(x) = – , x  0. Prove that lim f(x) = –, lim f(x) = .
x x + x 0–
0
5. Show that for the function f: R  R defined by
2
f(x) = x ,
f(x) exists for every a  R.
6. Find

3
+
-
(2x 3) (3x 2) 2
(i) lim 5
x w x + 5
-
3
(x + 1) 1 3
(ii) lim .
+
x w x 1
-
ì 2x for 0 s x < 1
ï
7. If g(x) = 4 for x = 1
í
ï
-
x
î 5 3x for 1 <  2.
find lim g(x)
x 1
8. Find
-
x - 2 æ x 1ö x
(i) lim (v) lim ç ÷
x 1 ø
x 4 x 4- x  è +
2
-
3x - x 10 æ sin 2xö 1 + x
(ii) Find lim 2 (vi) lim ç ÷
-
0
x 2 x + 5x 14 x è x ø
x 2
+
-
1 cosx æ x 1 ö
(iii) lim 2 (vii) lim ç ÷
+
x 0 x x è 2x 1 ø
-
sin x sina
(iv) lim
-
x a x a
Answers: Self Assessment
1. 1675 2. domain
3.  –  approach 4. |f(x) – A| <  for a – 6 < x < a

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