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Question 1

For the following function, , evaluate the following limits:

Question 2

Consider the piecewise function

- a) Graph the function on the domain [-3, 3].

- b) Find the .

- c) Find the .

- d) Find the .

- e) Find .

Question 3

Sketch a possible graph of a function that has domain , has zeros (intersects the x-axis) at -2, and 1, and the , and

Possible answer:

Question 4

Sketch a possible graph of a function that has domain , but not , goes through the points (-4,2) and (0,2), and has .

Possible answer:

Question 5

Sketch a possible graph of a function that has zeros of -2, and 1, and , but , and

Possible answer:

Question 6

Estimate the value of by trying values very close to 0 (from both the right and the left, obviously, since it's not a one-sided limit). Once you've made a reasonable estimate, check it by actually graphing the function near that point.

Note: The limit value converges to "e"

Question 7

Estimate the value of by trying values very close to 2 (from the left only, of course). Once you've made a reasonable estimate, check it by actually graphing the function near that point.

Estimation:

Graph:

Question 8

For the function , approximate the slope of the tangent line at (1,3) by doing the following:

- Make a graph of f(x) and plot a point at , somewhere nearby the point (1,3).

- Write the slope of the secant line that joins the point (1,3) and the point (a, f(a)) in terms of a, and simplify.

- Take the to determine the actual slope of the tangent line at (1,3).

Question 9

For the function , approximate the slope of the tangent line at (1,- 2) using the secant line approximation method (see question ID 158 for the individual steps).

Question 10

Evaluate the following limit:

Question 11

Evaluate this limit:

Question 12

Evaluate the following limit:

Question 13

Evaluate this limit:

Question 14

Evaluate this limit:

Question 15

Evaluate the following limit:

Question 16

AP Prep:

Which of the following limits exist?

Question 17

AP Prep:

Consider the function:

Which of the following are true?

- exists.

- exists.

Question 18

[Try This!]:

What is the formal definition of a limit of a real function? This is also known as the epsilon-delta definition of a limit (hint: wikipedia!). Once you've got the definition, try to figure out what it means, and see if you can use it to show that .

Question 19

[Try This!]:

What is the formal definition of a limit of a real function? This is also known as the epsilon-delta definition of a limit (hint: wikipedia!). Once you've got the definition, try to figure out what it means, and see if you can use it to show that .

Question 20

[Try This!]:

is equivalent to:

(a) For all , there is an such that if , then .

(b) For all , there is a such that if , then .

(c) For all , there is a such that if , then .

(d) For all , there is an such that if , then .

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The answers are available online within StudyForge. Click on the question box to expand it and press the "check answer" button.