How do you use the definition of continuous to prove that f is continuous at 2 given #f(x) = x^2 -3x +5#?
The answer depends on whether you are allowed to use limit laws or not. If you are, then use them; if not, then use "epsilons and deltas".
If you are allowed to use limit laws, such as the fact that
Now use the facts (assuming you allowed to) that
On the other hand, if you are not allowed to use limit laws, then you need to use an epsilon/delta proof. That is, we must show that given any number
Towards this end, note that
If we make this assumption, then
Now we are ready for the proof (finally, believe it or not). Here goes:
Also, by factoring and using properties of absolute value,
We're done, this proves that
"Show" is a writing assignment. See explanation.
(When you are asked to "Show" or "Prove" or "Verify" a statement, that is a writing assignment in a math class.
Your job is to write something to convince a reader that the statement is true.)
The first step is to recall the definition of continuous. Some variations are possible, but in your class or textbook it will be some version of:
Using the definition to show that this function
We start by pointing out that
Now we need to convince our reader that:
The way you do this will depend on both your knowledge and your assumed audience's knowledge. I am going to giess that you (and your audience) have learned the properties of limits (sometimes called the limit laws) and I'll use those:
# = lim_(xrarr2)x^2-lim_(xrarr2)3x+lim_(xrarr2)5" "#(sum and difference property)
# = (lim_(xrarr2)x)^2-3lim_(xrarr2)x+lim_(xrarr2)5" "#(power and constant multiple)
# = (2)^2-3(2)+5" "#(limt of #x#and limit of a constant)
# = 4-6+5 = 3" "#(arithmetic)
So we have shown that for