Question

Show that the function f(x) = 1 - sqroot(1 - x^2) is continuous on the interval [-1, 1].

Answer 1

what are you supposed to use? pretty much any function you can write down as an equation is continuous on its domain, and the domain of this function is \([-1,1]\)

Answer 2

I am honestly not too sure.. the textbook just substituted x with a according to Limit laws and came up to the conclusion that its continuous but it doesn't make too much sense to me

Answer 3

maybe that the limit of the square root is the square root of the limit, so as long as \(a\in [-1,1]\) you have
\[\lim_{x\to a}1-\sqrt{1-x^2}=1-\sqrt{\lim_{x\to a}(1-x^2)}\]and now the limit of the square is the square of the limits so get
\[=1-\sqrt{1-\lim_{x\to a}x^2}=1-\sqrt{1-a^2}\]

Answer 4

it is just using the so called "limit laws" so if you have
\[\lim_{x\to a}f(x)=L\] then
\[\lim_{x\to a}\sqrt{f(x)}=\sqrt{\lim_{x\to a}f(x)}=\sqrt{L}\]

Answer 5

Oh ic.. so its basically replacing any x with a according to the limit laws.. and I do this when they ask to show that a function is continuous?

Answer 6

hope it is clear what job you are supposed to do
to show that a function is continuous \(f\) at a point \(a\) you need to show that
\[\lim_{x\to a}f(x)=f(a)\]
in your case you have to show that
\[\lim_{x\to a}1-\sqrt{1-x^2}=1-\sqrt{1-a^2}\]

Answer 7

i meant to write

to show that a function \(f\) is continuous at a point \(a\) you need to show that...

Answer 8

yes so to write it all out you bring the limit inside the square root, because the limit of the square root is the square root of the limit
then once it is inside the square root, you bring it inside the square as well, since it is obvious that \[\lim_{x\to a}x=x\] you then can say that \[\lim_{x\to a}x^2=a^2\]

Answer 9

you will have to admit that it is all rather silly
as i said, pretty much any function you can write downs as a formula is continuous on its domain
so for example
\[f(x)=\frac{x}{\sqrt{x-1}}\] is continuous on
\[(1,\infty)\]

Answer 10

Ohh I get it now, thanks a lot for the help I really appreciate it.

Answer 11

yw