Question

examine the continuity of f defined by
f(x)= cos x + sin x , if x < 0
x+1 , if x >= 0

Answer 1

so you need to find the limit as x->0
for both the functions
can u ?

Answer 2

cos x + sin x , if x < 0 i don't understand how i should go about it

Answer 3

just find the limit
\(\lim \limits_{x \to 0} \cos x +\sin x\)

Answer 4

is it like this lim x->0 cos x + lim x->0 sin x ?

Answer 5

1+0 = 1

Answer 6

yes,
now find the limit of x+1 , as x ->0

Answer 7

but don't we have to find it for lim x-> 0- and lim x-> 0+

Answer 8

\(\lim \limits_{x \to 0^-} f(x) \)


\(\to 0^-\) mean very near to 0 but LESS than 0
so we select the function cos x + sin x for it
as it is with x< 0

Answer 9

\(\lim \limits_{x \to 0^+} f(x)\)

\(\to 0^+\) means very near to 0 but GREATER than 0
so we select f(x) = x+1 for this limit as its with x >=0

Answer 10

and for the limit \(\to 0\) to EXIST, these two individual limits MUST be EQUAL

Answer 11

so for lim x-> 0- will cos x + sin x = 1+0 ?

Answer 12

yes, so lim \(x \to 0^- =1\)
how about
\(\lim x\to0^+\)

Answer 13

1

Answer 14

so lim x-> 0 exists

Answer 15

yes and it = 1

Answer 16

so it is continuous across all R

Answer 17

thats correct :)

Answer 18

i don't understand this part cos 0 is 1 but why should lim x-> o- cos x be 1 ?

Answer 19

and lim x-> o- sin x be 0 ... shouldn't they be having some other values

Answer 20

for the left hand limit (or right hand limit), we solve it like a normal limit question,
\(\Large \lim \limits_{x \to a^-}f(x) = f(a)\)

Answer 21

ok thanks :)

Answer 22

its like x is VERY VERY NEAR to 0 ,
but the - *minus* just suggest that x is LESS than 0