Question

MATHEMATICS-IX)
vr. (716+Vx+4
+ x - 16
/16 - Ja
4 = 4 + 4 -
= Lim
= Lim
16+
Also,
f10) = 0
Now, f(x) will be continuous at x = 0 if Lim f (x) = Lim f (x) = f (0)
8=8=a
> a = 8
EXAMPLE 17. Determine the values of a, b, e for which the following function is
continuous at x=0:
sin (a + 1) x + sin x
for aco
f(x)=
for x=0
[AICBSE 2016]
Vx+bx² ſi
for x>0
X
Solution. At x = 0:
sin (a +11x + sin x
L.H.L = Lim
sin (a +1(h) + sin(-h)
(Putting x = 0 -hu-h)
Lim
sin (a +1) h
sinh
sin (a + 1) h - sinh​

Answer 1

Answer:

In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval [a, b], then it takes on any given value between f(a) and f(b) at some point within the interval.

This has two important corollaries:

If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem).[1]

The image of a continuous function over an interval is itself an interval