Question

Discuss the continuity of SIN function.Please explain!

Answer 1

think of this as a thumb rule
if the differntial of the given function is defined every where then the given function is continuous every where
differentiating sinx you get cosx which is defined for every x
so SIN is continuous every where

Answer 2

Can we not prove it mathematically,if yes then how?

Answer 3

You have to show that for any \(\varepsilon>0\), there is a \(\delta>0\) such that \(|x-c|<\delta\) implies \(|\sin x-\sin c|<\varepsilon\).

Answer 4

this is very tough which u have replied, I can't get through it.please tell in easy way

Answer 5

for checking where would function be discountous think of for some asymptoes if there is any then it is discontinuous like TAN is discontinuos when COS become ) .Now when does cos become 0 these will be your retricemptoes

Answer 6

*when COS become 0

Answer 7

k!

Answer 8

This question is very subjective: you can say that since \(\sin\) is differentiable, it must be continuous.

Answer 9

But we can't figure out any for sinx
try it for others you will get
Tell me when will SEC be discontinuous

Answer 10

may I send u solution, then will u make me understand that?

Answer 11

No ...... thats not true as TAN is differentiable bt not for every R so U should say if it is differentiable in R

Answer 12

\(\mathbb{R}\) is always the universe of discourse... unless otherwise specified...

Answer 13

sol. is as follows-

Answer 14

to see this we use the following facts-
\[\lim_{x \rightarrow o} \sin x=0\]
we have not proved it,but is intuitively clear from the graph of sinx near zero.

Answer 15

now , observe that f(x) =sinx is defined for every real no.. Let 'c' be a real no..Put x=c+h.
If x->c we know that h->0.thus,

Answer 16

\[\lim_{x \rightarrow c} f(x)=\lim_{x \rightarrow c} \sin x\]

Answer 17

u can prove it if u want to
sin x=x-x^3/fact3+x^5/fact5........
when u put x=0 u get sinx =0

Answer 18

\[\lim_{h \rightarrow 0} \sin (c+h).\]

Answer 19

\[\lim_{h \rightarrow 0} [\sin c \cos h+ \cos c \sin h]\]

Answer 20

yes this can be used to understand continuity
if in neighbourhood of x=c sinx=sinc
ie at x=0 and at x=0- and x=0+
where 0+ and 0- is the neighbourhood of c(here c=0)

Answer 21

previous one

Answer 22

\[\lim_{h \rightarrow 0} [\sin c \cos h]+\lim_{h \rightarrow 0} [\cos c \sin h].\]

Answer 23

=sin c+0=sin c=f(c)

Answer 24

ya.....

Answer 25

Thus, \[\lim_{x \rightarrow c} f(x)=f(c)\] & hence f is a continuous function.

Answer 26

hmmm

Answer 27

did u understand the solution.

Answer 28

yeess I do

Answer 29

ok!