What Can the Limit of a Function Tell You About the Behavior of the Graph?

What can the limit of a function tell you about the behavior of the graph?

Evaluate each limit if it exists. If the limit DNE, what can you conclude about the behavior of the graph?

What key features are relevant to the function being analyzed?

State the key features of each function. Use the key features to sketch the graph.

1)

2)

3)

4)

5)

6)

7)
/ 8)

9)

10)

11)

How do the key features of a rational function connect to real-world applications?

Part 3 #1

When distance and time are held constant, the average rates, in miles per hour, during a round trip can be modeled by , where represents the average rate during the first leg of the trip and represents the average rate during the return trip. Sketch the graph of and interpret the key features in the context of the problem.

The concentration of a certain medicine in a person’s body after hours is modeled by . Interpret the graph and the key features of the function in the context of the problem.

The business plan for a new car wash projects that profits in thousands of dollars will be modeled by the function , where , is the week of operation. Interpret the graph and key features of the function in the context of the problem.

Answer Key:

What can the limit of a function tell you about the behavior of the graph?

The limit of f(x) at x=2 approaches -4.The limit of f(x) at x=2 DNE; V.A.

The limit of f(x) at x=1 approaches -1.

The limit of f(x) at x=-1 DNE; point of discontinuity at (-1,-5).

Graph 1
(d) -∞
(e) DNE
(f) 2
(g) DNE
(h) ∞ / Graph 2
(d) DNE
(e) 0
(f) -∞
(g) 1

What key features are relevant to the function being analyzed?

State the key features of each function. Use the key features to sketch the graph.

1)


2)


Hole at x = -3
3)


6)

7)

8)
Hole at x = 3
/ 4)


5)


9)

Slant Asymptote, No HA.
NumDenom
10)


11)

Answer Key

Part III #1

Function is most likely designed to represent the trip when r1 > 30, as any rate r1<30 would result in a negative rate.

It’s a comparison of rates, so after 30 mph, the faster you drive on the first leg of the trip, the slower the return trip is till it approaches a minimum return speed of 30 MPH.

#2)

Concentration begins at Zero. Hits a max around 3 hours, and as the body begins to process it, slowly disappears from the body. Frighteningly, trace amounts of the medicine remains in the body long after its original dose and will never fully dissipate from the body.

#3)

Used to represent a function’s profit from week 0. In its 1st week it is in debt. It breaks even after the 1st week (great project). Profits continue to increase until it reaches an equilibrium where the profits settle at just under $1000 dollars.