unit 1 differential calculus - IGNOU

SAQ 4
lim
=
x →0
Evaluate the limits that exist :
(i)
(iii)
(v)
(vii)
(ix)
lim
x
2
− x − 6
x →− 2 2
( x 2)
2
( x
lim
+
x →3 x −
2
x − 2x
lim
x →0
sin 3x
+ x − 12)
3
sin ( x − a)
lim
x →a
2
( x − a)
sin α
lim
x →0 sin β
x
x
1.5 CONTINUITY
.
⎛ x ⎞
⎜ ⎟
( 2)
(cos x)
⎜ 2 ⎟
⎜ sin x ⎟
⎜ ⎟
⎝ 2 ⎠
= (2) (1) (1) = 2.
2
(ii) ⎟ ⎛ 3x
8 ⎞
lim ⎜ +
x →− 4 ⎝ x + 4 x + 4 ⎠
(iv)
(vi)
(viii)
lim
x →0
2
tan x
x
2
sin x
lim
x → x(
1 − cos
lim
x →0
0 x
3 + x −
In ordinary language, to say that a certain process is “continuous” is to say that it
goes on without interruption and without abrupt changes. In mathematics,
continuity of a function can also be interpreted in a similar way.
Like limits, the idea of continuity is basic to calculus. First we introduce the idea
of continuity at a part (or number) a, and then about continuity on an interval.
Continuity at a Point
Definition 8
Let f be a function defined at least on an open interval (a – p, a + p).
We say f is continuous at a if, and only if,
or, equivalently,
lim f ( x)
= f ( a).
x →a
lim
h →0
f ( a + h)
= f ( a).
x
)
3
Differential Calculus
31