 # How do you prove that the function f(x) = x^2 -3x +5 is continuous at a =2? Zachariah Norris 2022-09-15 Answered
How do you prove that the function $f\left(x\right)={x}^{2}-3x+5$ is continuous at a =2?
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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 $\underset{x\to a}{lim}f\left(x\right)$ are defined and equal.
In our example:
$f\left(2\right)={2}^{2}-3\left(2\right)+5=4-6+5=3$
$={2}^{2}+4h+{h}^{2}-3\left(2\right)-3h+5$
$={2}^{2}+4h+{h}^{2}-3\left(2\right)-3h+5$
$=4-6+5+h+{h}^{2}$
$=3+h+{h}^{2}\to 3$ as $h\to 0$
So $\underset{x\to 2}{lim}f\left(x\right)=\underset{h\to 0}{lim}f\left(2+h\right)=3=f\left(2\right)$