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

Apr 5, 2016

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 $f \left(x\right)$ is continuous at a point $a$ if both $f \left(a\right)$ and ${\lim}_{x \to a} f \left(x\right)$ are defined and equal.

In our example:

$f \left(2\right) = {2}^{2} - 3 \left(2\right) + 5 = 4 - 6 + 5 = 3$

$f \left(2 + h\right) = {\left(2 + h\right)}^{2} - 3 \left(2 + h\right) + 5$

$= {2}^{2} + 4 h + {h}^{2} - 3 \left(2\right) - 3 h + 5$

$= 4 - 6 + 5 + h + {h}^{2}$

$= 3 + h + {h}^{2} \to 3$ as $h \to 0$

So ${\lim}_{x \to 2} f \left(x\right) = {\lim}_{h \to 0} f \left(2 + h\right) = 3 = f \left(2\right)$