CHAPTER - 13
LIMITS AND DERIVATIVES KEY POINTS
lim f(x) =
x c
lim x c
–
l
f(x) =
if and only if
lim f(x)
x c
lim where is a fixed real number.
x c
lim xn = cn, for all n N
x c
lim f(x) = f(c), where f(x) is a real polynomial in x.
x c
Algebra of limits Let f, g be two functions such that lim f(x) = x c
and lim g(x) = m, then
lim [ f(x)] = lim f(x)
x c
x c
=
l
l
for all R
lim [f(x) ± g(x)] = lim f(x) ± lim g(x) =
x c
x c
x c
l
lim [f(x).g(x)] = lim f(x). lim g(x) = l m
x c
x c
lim f(x ) f(x) x c lim = lim g(x ) x c g(x)
x c
l
m
, m 0 g(x) 0
x c
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± m
x c
1 1 1 = = provided lim f(x) l f(x)
lim x c
l
0 f(x) 0
xc
n lim [(f(x)]n = lim lim f(x) = x c x c
l n,
for all n N
Some important theorems on limits
lim x0
lim
x
xa
lim x0
lim
f(x) = n
x0
f(–x)
n
a = nan – 1 xa
sin x x =1 where x is measured in radians.
1 1 x
lim x sin
x
tan x cos x 1 Note that lim 1 x x x0 x 0
1 cos x 0 x x 0
li m
lim
lim
e
x
x0
x
lim x0
lim x 0
a
1 1 x
1 x
log e a
log(1 x) 1 x
lim 1 x
1x
e
x 0
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XI – Mathematics
Derivative of a function at any point
A function f is said to have a derivative at any point x if it is defined in f(x h) f(x ) some neighbourhood of the point x and lim exists. h h 0 The value of this limit is called the derivative of f at any point x and is denoted by f'(x) i.e.
f '(x ) lim h 0
f(x h) f(x ) h
Algebra of derivatives :
d d cf x = c. f x where c is a constant dx dx
d d d f x g(x ) = f x g(x ) dx dx dx
d d d f x .g .g x = f x . g x g x f x dx dx dx
d dx
d d g x f x f x . g x f( x) dx dx 2 (x) = g(x) g x
If y = f(x) is a given curve then slope of the tangent to the curve at the point (h, k) is given by
dy and is denoted by ‘m’. dx h,k
VERY SHORT ANSWER TYPE QUESTIONS (1 MARK) Evaluate the following Limits :
1.
lim x3
2.
lim x 0
2x 3 x 3 sin 3x x
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76
3.
lim
x
x0
4.
lim x x2
2
ta n
2
3x
2
5x 1
Differentiate the following functions with respect to x : 5.
x 2 2 x
6.
x2 tanx
7.
x si n x
8.
l og x x
9.
2x
10 .
If f(x) = x2 – 5x + 7, find f'(3)
11 11..
If y = sinx sinx + tan tanx, x, find find
dy at x dx 3
SHORT ANSWER TYPE QUESTIONS (4 MARKS) 12.
5x 4, 0 x 1 , I f f x show that lim f(x) exists. 3 x 1 4x 3x , 1 x 2
13.
x x , x 0 , show that lim f(x) does not exist. I f f x x x 0 x 0 2 ,
14.
Lett ff(x Le (x)) be a func functi tion on defi define ned d by by
4x 5, If x 2, , Find , if lim f(x) exists x 2 , If x 2, x
f x
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XI – Mathematics
Evaluate the following Limits :
15.
lim x3
16.
x x
lim
4x 3 2x 3 2
x
lim
1 x
x0
18.
2
2x x
lim x 0
17.
2
x
xa
x
5 7 2 7
1 x
a a
5 7 2 7 5
19.
lim
x
xa
5
2 2 x
a
a
20.
1 cos 2mx x 0 1 cos 2nx
21.
lim
tan x sin x x
3
22.
x tan x x 0 1 cos x
23.
lim
lim
x a
lim x a
25.
26.
2 2
lim
x0
24.
lim
s in x s i n a x a cos x cos a co t x co t a 1 sec
lim
x
2
tan x
x
x0
3
e
x
e x
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x
78
27.
x 1 x 1 l og e x
28.
lim
lim
x e
29.
lim x4
30.
31.
log x 1 xe 3
5 x
1
5x
a 2x 2x
lim
3a x 2 x
x a
lim x 0
3x
sin(2 x ) sin(2 x) x
Differentiate the following functions with respect to x from first principle: 32.
33.
2x 3
x
2
1 x
34 .
ex
35 .
log x
36 .
cosec x
37 .
cot x
38 .
ax
Differentiate the following functions with respect to x : 39.
(3 x
1) (2 (2 x
1)
x
40.
3 x 1 x
41.
x 1 x 2 1 2 x x
42.
sin x x cos x x sin x cos x
43 .
x3 ex sinx
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XI – Mathematics
44 .
xn logax ex
45.
e
46.
1 log x 1 log x
47 .
ex sinx + xn cosx
48.
If y
x
49.
If y
1 cos 2x 1 cos 2x
50.
If y
x a
x
log x sin x
2xy 51 51..
1 x
, prove that 2x
find
dy y 2 x dx
dy dx
a , prove that x
dy x a dx a x
For th the ec cur urve ve f(x) f(x)=( =(x x2 + 6x–5)(1–x), find the slope of of the tangent at x = 3.
LONG ANSWER TYPE QUESTIONS (6 MARKS) Differentiate the following functions with respect to x from first principle: 52.
cos x x
53 .
x2 sinx
Evaluate the following limits :
54.
55.
lim x
6
lim x 0
2 sin 2 sin
2
2
x sin x 1
x 3 sin x 1
cos 2x cos 3x cos 4x 1
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80
ANSWERS 1.
1 2
2.
3
3.
9
4.
–5
5.
1 2 2 2 x
6.
2 x tanx + x2 sec2x
7.
cosecx – x cotx cosecx
8.
0
9.
2x loge2
10.
1
11.
9 2
14.
= –1
15.
1 2
16.
1 2 2 3
17 .
1
19.
3 5 a 2 2 2
20.
21.
1 2
22.
2
23 .
cosa
24.
sin3a
25.
26.
2
27 .
1
28.
1 e
29.
31 .
18.
5 7 a 2
3 2
1 3
2 cos2
30.
32.
81
m n
2 2
2 3 3 1 2x 3
XI – Mathematics
33.
x
2
1
x
2
34 .
ex
35.
1 x
36.
– cosecx. cotx
37 .
– cosec2x
38 .
a
1
39.
41.
6
3x
2
3 2 x 2 1
x
2
40.
3
1
x
42.
4
44 .
exxn
45.
x e 1 sin x e x log x cos x x sin
3
x
5/ 2
9 2
x
2
sin x x si
cos x
2
2
x
2 46.
x 1 log x
49 .
sec2x
54 .
2
logax + loga + x logax}
2
x sin x cos x x
3x
2x
x2ex (3 sinx + x sinx + x cosx)
52.
logea
3
1 2 x 2
43 .
– 1{n
x
2
–3
XI – Mathematics
47.
1 x e 1 x log x x
51 .
–4 6
53 .
2x sinx + x2 cosx
55 .
82
5 16
