100 points!!
Answer with steps in detail please.

For number 22.
Determine whether the following function is continuous at a. Use the continuity checklist to justify your answer.

The function is continuous at x=3

Step-by-step explanation:

Continuity of a Function

There are three conditions a function f(x) must comply to be continuous at x=a:

Should one of the conditions fail, the function would not be continuous at x=a.

Let's test the continuity of the function at x=3

First, we find out if f(3) exists. The second condition of the piecewise function indicates that f(3) = 2

Now let's compute

This is an indeterminate limit which must be simplified by factoring the numerator:

Simplifying

The limit exists and is equal to 2

The third condition demands that the function and the limit are equal. We can see this condition is met, thus we conclude the function is continuous at x=3

Yes, continuous

Step-by-step explanation:

From the left:

x = 2.9999

(2.9999² - 4(2.9999) + 3)/(2.9999 - 3)

= 1.9999

From right:

x = 3.0001

(3.0001² - 4(3.0001) + 3)/(3.0001 - 3)

= 2.0001

From both sides, limit approaches 2

Which is also the value at x = 2

18.75 gallons

step-by-step explanation: