# Thread: Prove continuity at a point using epsilon/delta

1. ## Prove continuity at a point using epsilon/delta

Hi,
I need to prove that f(x) = cuberoot[x+3] is continuous at x = -2 (using: u^3 - v^3 = (u-v)(u^2 + uv + v^2). I know that I need to manipulate mod(f(x) - f(-2)) to be less than epsilon for some delta but I'm struggling with the algebra.
Any help would be greatly appreciated!

2. Originally Posted by s0791264
Hi,
I need to prove that f(x) = cuberoot[x+3] is continuous at x = -2 (using: u^3 - v^3 = (u-v)(u^2 + uv + v^2). I know that I need to manipulate mod(f(x) - f(-2)) to be less than epsilon for some delta but I'm struggling with the algebra.
Any help would be greatly appreciated!
$f(x) = \sqrt[3](x+3)$ at x = -2

what you want to show is the following:

for $\epsilon > 0$, there exists $\delta > 0$ such that:

$|f(x) - f(c)| < \epsilon$ for all x that belongs to dom f\{c} s.t. $|x-c| < \delta$

i remember working backward in problem like these. Start out with something like this...

let $\epsilon > 0$,

for $|x+2| < \delta$

$|\sqrt[3](x+3) - \sqrt[3](-2+3)| = |\sqrt[3](x+3) - 1|$

... (you'll prolly need to use some algebraic formula for cube/cluberoots and perhaps triangle inequality as well
...
...

$< \epsilon$

P.S. I'll try this later on.
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