define f:R-R by f(x)=x^2 +3x -5. use the continuous function definition to prove that f is continuous at 3

Question

define f:R->R by f(x)=x^2 +3x -5. use the continuous function definition to prove that f is continuous at 3

anonymous · Answer

Are we supposed to use the $\large \epsilon-\delta$ definition or the limit definition of a continuous function?

anonymous · Answer

Let f:D->R and let c be an element of D then f is continuous at c if for every epsilon>0 there exists a delta>0 such that If(x)-f(c)I

anonymous · Answer

Use the fact that
$$
f(x)-f(3)=x^2+3 x-18=(x-3) (x+6)
$$

anonymous · Answer

Let us suppose that  x< 4, so
$$
|f(x)-f(3)|=|(x-3) (x+6)| < 10|x-3|
$$

anonymous · Answer

can I make x less than any number? or does it have to be 4?

anonymous · Answer

Take $ \delta < \frac {\epsilon }{10} $
If $$
|x-3| < \delta  \implies |f(x)-f(3)| < 10|x-3| < 10 \frac {\epsilon}{10}=\epsilon

$$

anonymous · Answer

Your x is going to be near 3 so I took a number a bit bigger than it.

anonymous · Answer

oh so i could also take 2 and make delta<epsilon/8?

anonymous · Answer

To be precise one should take |x| < 4 and $ | x-3| <\frac{ \epsilon}{10}$

anonymous · Answer

I never know what value to suppose while writing proofs

anonymous · Answer

But thank you so much..