Problem 1:  Graphing a Function (18 points)

Sketch the graph of a function f satisfying all of the following properties.

a)  (2 points)  f is an odd function with domain [−6,6].

b)  (3 points)  f has intervals of continuity: [−6,−4), [−4,−2), (−2,0), (0,2), (2,4], and (4,6].

c)  (3 points)  f (0) = 0, f (4) = 0 and f (6) = 3.

d)  (3 points)   lim  f (x) = 4 and  lim f (x) = 1 x →0+                                                x →2

e)  (1 point)   lim  f (x) = −2 .

f)  (3 points)  f (x) > 0 for x in (0,2) ∪ (2,4) ∪ (5,6].

g)  (3 points)  f (x) < 0 for x in (4,5) and for x = 2.

Problem 2:  Using Limit Laws and the Squeeze Theorem (14 points)

Let g be the function displayed in the graph below.

a)  Limit laws are used to help evaluate almost every limit we’ll see this term. Often we do not mention them explicitly, but it is important that you remember how they are used. For example, we can evaluate  lim 2xeg(x)  using the following steps:

x →4

x(l) 2xeg(x)   =  (x(l) 2x)(x(l) eg(x)) = (2 ∗ 4) (ex(l)g(x)) = (8) (e3 )

Answer the following questions to explain why this process works.  For each step you should cite relevant limit laws or facts about g(x)

i)  (2 points) Why can we say:x(l) 2xeg(x)   =  (x(l) 2x)(x(l) eg(x))?

ii)  (2 points) Why can we say:  (x(l) 2x)(x(l) eg(x)) = (2 ∗ 4) (ex(l)g(x))?

(               )

b)  (8 points) Suppose that for all values of x near x = 2, the function s(x) satisfies the inequality log3 (4x + 1) ≤ s(x) ≤ g(x)

Evaluate the following limit, and EXPLAIN your answer.  Explanations should include finding relevant limits and properly citing any theorems you use.

lim s(x)

x →2

Problem 3:  Exploring the 0/0 Form (10 points)

a) A key technique in evaluating limits of the form 0/0 is ”factoring and canceling”. Lets explore why this technique works with a simple example.

3x(x − 2)                                                             3x

2(x − 2)                                                              2 .

ii)  (2 points) These two different functions have the same limit as x approaches 2. That is,

3x(x − 2)               3x

x →2   2(x − 2)        x →2   2 .

In your own words, explain why these two limits are equal, despite the functions being different.

For the following limit, write the form in the space provided.  Then evaluate the limit if it exists.  If the limit does not exist, explain why.  In this part, you do not need to cite individual limit laws that are used.  Otherwise, justify your answers by showing your work. Points will be taken off for skipped steps or improper notation .

x2  − 1    

x → −1 x2  − 2x − 3

Problem 4:  Limits and Continuity for a Piecewise Function (18 points)

For this problem, we will use all of our techniques to explore the continuity properties of the piecewise function

 x2 + 2x − 8

      x − 2

' ' ' ' ' ' '

'(3^x + 2

if x < 2

if x = 2

if x > 2