# How do you use the definition of continuous to prove that f is continuous at 2 given #f(x) = x^2 -3x +5#?

##### 2 Answers

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".

#### Explanation:

A function

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:

Let

Also, by factoring and using properties of absolute value,

We're done, this proves that

"Show" is a writing assignment. See explanation.

#### 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:

A function **if and only if**

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)

That is:

So we have shown that for