Question

Find the limit as x approaches 0 of:
x^2 + cosx

Answer 1

Did you plug in 0? :)
This function is continuous at x=0 so you may do so

Answer 2

Why do you do that?

Answer 3

because it is easiest

Answer 4

if \(f\) is continuous then \(\lim_{x\to a}f(x)=f(a)\)

Answer 5

Okay, but is there another way to do this? Like with identities?

Answer 6

no

Answer 7

How do you know it is continuous?

Answer 8

f(0) exists

Answer 9

and the functions is continuous everywhere

Answer 10

because x^2 is continuous everywhere
and cos(x) is continuous everywhere

Answer 11

every function you know that is definded on an interval is continous on its domain

Answer 12

sine, cosine, exp, log, any polynomial, any rationa function etc all continuous on their domains

Answer 13

okay but if it were something like sinx/x it would not be continuous because x is undefined at 0?

Answer 14

even that one is continous on its doman. just happens that the domain does not include zero

Answer 15

limit exists at x = 0, but the function does not, so it is not continous at 0
in order for a functioni to be continuous at a point "a" it must be defined at "a"

Answer 16

okay thanks:)