Ap Calculus Essay

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AP Calculus
Review
Limits, Continuity, and the
Definition of the Derivative
Teacher Packet

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The College Board was not involved in the production of, and does not endorse, this product.
Copyright © 2008 Laying the Foundation, Inc., Dallas, Texas. All rights reserved.
These materials may be used for face-to-face teaching with students only.

Limits, Continuity, and the Definition of the Derivative
Page 1 of 18

DEFINITION

Derivative of a Function

The derivative of the function f with respect to the variable x is the function f ′ whose value at x is f ′( x ) = lim

h→ 0

f ( x + h) − f ( x) h Y

(x+h, f(x+h))
(x, f(x))

X

provided the limit exists.
3 + x5
=∞
x → ∞ x 2 − 3x + 7 lim ⎛ x5 ⎞
This function has end behavior like x ⎜ 2 ⎟ . The function does not reach a limit, but
⎝x ⎠ to say the limit equals infinity gives a very good picture of the behavior.
3

If the x with the largest exponent is the same, numerator and denominator, the limit is the coefficients of the two x’s with that largest exponent.
3 + 4 x5
4
lim
= . As x → ∞ , those x 5 terms are like gymnasiums full of sand. x → ∞ 7 x 5 − 3x + 7
7
The few grains of sand in the rest of the function do not greatly affect the behavior of the function as x → ∞ .

Copyright © 2008 Laying the Foundation, Inc., Dallas, Texas. All rights reserved.
These materials may be used for face-to-face teaching with students only.

Limits, Continuity, and the Definition of the Derivative
Page 5 of 18

LIMITS

lim f ( x ) = L x→ c

The limit of f of x as x approaches c equals L.
As x gets closer and closer to some number c (but does not equal c), the value of the function gets closer and closer (and may equal) some value L.

One-sided Limits lim f ( x ) = L

x→ c−

The limit of f of x as x approaches c from the left equals L. lim f ( x ) = L

x→
lim ⎜ x ⎟ = x→ ∞
⎝ e ⎠

⎛ x2 − 9 ⎞
6. lim ⎜
⎟=
x→ −∞ ⎜ 2x − 3 ⎟
⎝
⎠
⎛ x2 − 9 ⎞
7. lim ⎜
⎟=
x→ ∞ ⎜ 2x − 3 ⎟
⎝
⎠

Copyright © 2008 Laying the Foundation, Inc., Dallas, Texas. All rights reserved.
These materials may be used for face-to-face teaching with students only.

Limits, Continuity, and the Definition of the Derivative
Page 7 of 18

Practice Problems
Limit as x approaches a number
8. lim ( x 3 − x + 1) x→ 2

⎛ x2 − 4 ⎞
9. lim ⎜
⎟=
x→ 2
⎝ x−2 ⎠

⎛ 3 ⎞
10. lim− ⎜
⎟=
x→ 2 ⎝ x − 2 ⎠

⎛ 3 ⎞
11. lim+ ⎜
⎟=
x→ 2 ⎝ x − 2 ⎠

⎛ 3 ⎞
12. lim ⎜
⎟=
x→ 2 x − 2
⎝
⎠

⎛ 3 ⎞
13. lim+ ⎜
⎟=
x→ 2 ⎝ 2 − x ⎠

⎛ sin x ⎞
14. limπ ⎜
⎟=
x ⎠ x→ ⎝
4

⎛ tan x ⎞
15. limπ ⎜
⎟=
x ⎠ x→ ⎝
4

Copyright © 2008 Laying the Foundation, Inc., Dallas, Texas. All rights reserved.
These materials may be used for face-to-face teaching with students only.

Limits, Continuity, and the Definition of the Derivative
Page 8 of 18

1.

sin ( x + h ) − sin ( x )
?
h→0 h What is lim

(C) − sin x

(A) sin x

(B) cos x

(D) − cos x

(E) The limit does not exist

⎛π
⎞
⎛π ⎞ cos ⎜ + Δx ⎟ − cos ⎜ ⎟
⎝3
⎠
⎝ 3⎠ =
2. lim
Δx → 0
Δx
(A) −

(D)

3. lim

3
2

(B) −

1
2

(E)

( x + h)

h→0

3

h

− ( x3 )

1
2

(C) 0

3
2

=

(A) − x 3

(B) −3x 2

(D) x