Question

How do you prove that the function `f(x) = x^2 -3x +5` is continuous at a =2?

Answer 1

Prove using limit definition of continuity...

Explanation:

Any polynomial function is continuous everywhere, but let's prove this particular example using the limit definition of continuity...

A function `f(x)` is continuous at a point `a` if both `f(a)` and `lim_(x->a) f(x)` are defined and equal.

In our example:

`f(2) = 2^2-3(2)+5 = 4-6+5 = 3`

`f(2+h) = (2+h)^2-3(2+h)+5`

`= 2^2+4h+h^2-3(2)-3h+5`

`= 4-6+5+h+h^2`

`= 3+h+h^2 -> 3` as `h->0`

So `lim_(x->2) f(x) = lim_(h->0) f(2+h) = 3 = f(2)`