Thread: show function is continuous

hi, the only instruction is to show the function is continuous. i know the epsilon delta definition, but when that is required, usually it is explicitly mentioned. so, i am thinking i need to use a different idea. the function is defined as: arcsin(x)/x when x=/=0, and 1 when x=0. so, i am thinking about what method to use. should i show that when arcsin(x)/x approaches 0, the limit is 1? is this a sufficient condition? should i show that arcsinx/x is continuous elsewhere? or can i just state this (and possibly why?)

You have to show :

right. so. how can i do this?

Originally Posted by learning

right. so. how can i do this?

L'Hopital's rule

is l'opital essential here, i will be showing what with this? i understand the conditions are met for it, but, thn what? lopital shows differentiabiliy, which is the next question. so i need to show continuity here in a way which does not show differentiability.

i now have to show that it is in fact differentiable at 0. i am thinking again l opital here. but showing the conditions apply?

To be differential at a point, the function needs to be continuous at that point AND reasonably smooth. To be smooth, that means that the gradients don't make sudden jumps. So the derivative ALSO needs to be continuous at that point.

i need to show that (arcsinx/x - 1 )/ x goes to 0 as x goes to 0. it should be l ' hopitalable but arcsin0/0 - 1 has to equal 0 for that to be so. in the original function, the function is defined as 1 at x=0, can i use this here to show the condition for l' hopital? even though its technically a new function? can i plug in that earlier definition?

L'Hopital's rule is used for the limit of a fraction - when you get or , you can take the derivative of the numerator and denominator and try again. So you have arcsin(x) for the numerator and x for the denominator. If you plug in the derivatives and set x=0, you should get the answer.

- Hollywood

in terms of showing differentiability at 0. would l'hopital be appropriate again.

To show it's differentiable, you have a different limit to prove: .

- Hollywood

absolutely, i know, but then if you use l'opital here. my issue is, can you use the fact that arcsinx/x was defined as 1 at 0, to say that the numerator is 0 at h=0. (thus l' hopital is valid).? or since its a 'new' function you can't do this. l'hopital

i know what i have to do just not how to do it

You need to use L'Hopital's rule twice:









Since the limit exists, the function is differentiable.

- Hollywood

right, thats what i thought. but its the first line i wasnt sure about. whether i need to multiply through by h first, or if i could use the definition of the function to 'replace' the arcsinh/h with a 1 (since the function is defined as 1 at 0, therefore the function - 1 would be defined as 0 at 0). though., the net result is the same. thats actually what i did. to be honest. what you wrote above is exactly what i did. thanks for the confirmation.