Student Packet

Limits, Continuity, and the Definition of the Derivative Page 1 of 13

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. You will want to recognize this formula (a slope) and know that you need to take the f ( x + h) − f ( x ) . derivative of f ( x ) when you are asked to find lim h→ 0 h

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Student Packet

Limits, Continuity, and the Definition of the Derivative Page 2 of 13

DEFINITION (ALTERNATE)

Derivative at a Point

The derivative of the function f at the point x = a is the limit f ′(a ) = lim x→ a

f ( x) − f (a ) x−a

Y

(a, f(a)) (x, f(x))

X

provided the limit exists. This is the slope of a segment connecting two points that are very close together.

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Student Packet

Limits, Continuity, and the Definition of the Derivative Page 3 of 13

DEFINITION

Continuity

A function f is continuous at a number a if 1) f ( a ) is defined (a is in the domain of f ) 2) lim f ( x ) exists x→ a

3) lim f ( x ) = f ( a ) x→ a

A function is continuous at an x if the function has a value at that x, the function has a limit at that x, and the value and the limit are the same. Example: ⎧ x 2 + 3, x ≤ 2 Given f ( x) = ⎨ ⎩3 x + 2, x > 2

Is the function continuous at x = 2 ? f (2) = 7 lim f ( x) = 7 , but the lim+ f ( x) = 8

x → 2−

x→ 2

The function does not have a limit as x → 2 , therefore the function is not continuous at x = 2.

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Student Packet

Limits, Continuity, and the Definition of the Derivative Page 4 of 13

Limits as x approaches ∞

For rational functions, examine the x with the largest exponent, numerator and denominator. The x with the largest exponent will carry the weight of the function. If the x with the largest exponent is in the denominator, the denominator is growing faster as x → ∞ . Therefore, the limit is 0. 3+ x lim 4 =0 x → ∞ x − 3x + 7 If the x with the largest exponent is in the numerator, the numerator is growing faster as x → ∞ . The function behaves like the resulting function when you divide the x with the largest exponent in the numerator by the x with the largest exponent in the denominator.

3 + x5 =∞ x → ∞ x 2 − 3x + 7 lim

⎛ x5 ⎞ This function has end behavior like x 3 ⎜ 2 ⎟ . The function does not reach a limit, but ⎝x ⎠ to say the limit equals infinity gives a very good picture of the behavior.

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 = . As x → ∞ , those x 5 terms are like gymnasiums full of sand. The 5 x → ∞ 7 x − 3x + 7 7 few grains of sand in the rest of the function do not greatly affect the behavior of the function as x → ∞ . lim

Comparing Rates of Growth

Let f ( x) and g ( x) be positive for x sufficiently large. f ( x) = ∞ , then f ( x) grows faster than g ( x) . x →∞ g ( x ) g ( x) If lim = 0 , then g ( x) grows slower than f ( x) . x →∞ f ( x ) g ( x) If lim = L ≠ 0 , then f ( x) and g ( x) grow at the same rate. x →∞ f ( x ) If lim

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Student Packet

Limits, Continuity, and the Definition of the Derivative Page 5 of 13 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→ c+

The limit of f of x as x approaches c from the right equals L.

Using the graph above, evaluate the following: lim f ( x ) =

x → 1−

lim f ( x ) =

x → 1+

lim f ( x ) = x→ 1

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Student Packet

Limits, Continuity, and the Definition of the Derivative Page 6 of 13 Practice Problems Limit as x approaches infinity

3x − 7 ⎛ ⎞ 1. lim ⎜ 4 ⎟= x → ∞ 5 x − 8 x + 12 ⎝ ⎠

⎛ 3x 4 − 2 ⎞ 2. lim ⎜ 4 ⎟= x → ∞ 5x − 2 x + 1 ⎝ ⎠

⎛ ⎞ x6 − 2 3. lim ⎜ ⎟= x → ∞ 10 x 4 − 9 x + 8 ⎝ ⎠

⎛ ⎞ 7 x4 − 2 4. lim ⎜ ⎟= 3 4 x → ∞ 5 − 2 x − 14 x ⎝ ⎠

⎛ sin x ⎞ 5. lim ⎜ x ⎟ = x→ ∞ ⎝ e ⎠ ⎛ x2 − 9 ⎞ 6. lim ⎜ ⎟= x→ −∞ ⎜ 2x − 3 ⎟ ⎝ ⎠ ⎛ x2 − 9 ⎞ 7. lim ⎜ ⎟= x→ ∞ ⎜ 2x − 3 ⎟ ⎝ ⎠ Extra: Order these functions from slowest-growing to fastest-growing as x → ∞ . 4 x , x 2 + 1 , ln x , e x

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Student Packet

Limits, Continuity, and the Definition of the Derivative Page 7 of 13 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

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Student Packet

Limits, Continuity, and the Definition of the Derivative Page 8 of 13

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 3

(E) The limit does not exist

(C) 3x 2

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Student Packet

Limits, Continuity, and the Definition of the Derivative Page 9 of 13

4. The graph of y = f ( x) is shown above. lim

(A) 1

(B) 5

(C) 7

⎧ x 2 − 3x − 4 , ⎪ 5. If f ( x ) = ⎨ x + 1 ⎪⎩ 2, (A) −5

(B) 0

x ≠ −1

(( f ( x) ) − 3 f ( x) + 7 ) = 3

x→2

(D) 9

(E) Does not exist

, what is lim f ( x ) ? x → −1

x = −1 (C) 2

(D) 3

(E) Does not exist

1 2

(D) 2

(E) Does not exist

⎛ 2 x 6 − 5 x 3 + 10 ⎞ 6. lim ⎜ = 2 6 ⎟ x→ ∞ ⎝ 20 − 4 x − x ⎠

(A) −2

(B) −

1 2

(C)

⎛ 2 x 5 − 5 x 3 + 10 ⎞ 7. lim ⎜ = 2 6 ⎟ x→ ∞ ⎝ 20 − 4 x − x ⎠

(A) −2

(B) −

1 2

(C) 0

(D)

1 2

(E) 2

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Student Packet

Limits, Continuity, and the Definition of the Derivative Page 10 of 13

1 1 + ⎞ ⎛ 8. lim ⎜ 1 + e 2 x ⎟ = x→ ∞ ⎝ ⎠

1

(A) − ∞ 1

(D) 1 + e 2

9. lim+ x→ 3

(E) ∞

5 = 3− x

(A) −∞ (D)

(C) e 2

(B) 0

5 3

(B) −5

(C) 0

(E) ∞

⎛ ⎞ 1 5n 3 10. If lim ⎜ ⎟ = , then k = 3 x → ∞ 20 − 3n − kn ⎝ ⎠ 2

(A) −10

(B) −4

(C)

1 4

(D) 4

(E) 10

11. Which of the following is/are true about the function g if g ( x ) =

( x − 2)

2

x2 + x − 6

?

I. g is continuous at x = 2 II. The graph of g has a vertical asymptote at x = −3 III. The graph of g has a horizontal asymptote at y = 0

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) II and III only

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Student Packet

Limits, Continuity, and the Definition of the Derivative Page 11 of 13

π ⎧ ⎪sin x, x < 4 ⎪ π ⎪ 12. f ( x ) = ⎨cos x, x > 4 ⎪ π ⎪ ⎪⎩ tan x, x = 4 What is limπ f ( x ) ? x→

(A) − ∞

4

(B) 0

(C) 1

(D)

2 2

(E) ∞

⎛ x− a⎞ 13. lim ⎜ ⎟= x→ a x a − ⎝ ⎠

(A)

14. lim+ x→ 0

1 2 a

(B)

1 a

(C)

a

(D) 2 a

(E) Does not exist

(D) 1

(E) ∞

ln 2 x = 2x

(A) − ∞

(B) −1

(C) 0

⎧ x2 , 15. At x = 4 , the function given by h ( x ) = ⎨ ⎩4 x,

(A) (B) (C) (D) (E)

x≤4 x>4

is

Undefined Continuous but not differentiable Differentiable but not continuous Neither continuous nor differentiable Both continuous and differentiable

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Student Packet

Limits, Continuity, and the Definition of the Derivative Page 12 of 13

Free Response 1 Let h be the function defined by the following: ⎧⎪ x − 1 + 3, 1 ≤ x ≤ 2 h( x ) = ⎨ 2 ⎪⎩ ax − bx, x > 2 a and b are constants. (a) If a = −1 and b = − 4 , is h( x ) continuous for all x in [1, ∞ ] ? Justify your answer.

(b) Describe all values of a and b such that h is a continuous function over the interval [1, ∞ ] .

(c) The function h will be continuous and differentiable over the interval [1, ∞ ] for which values of a and b?

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Student Packet

Limits, Continuity, and the Definition of the Derivative Page 13 of 13

Free Response 2 (No calculator) x3 + 2 x 2 − 3x Given the function f ( x) = . 3x 2 + 3x − 6 (a) What are the zeros of f ( x) ? (b) What are the vertical asymptotes of f ( x) ? (c) The end behavior model of f ( x ) is the function g ( x) . What is g ( x) ? (d) What is lim f ( x) ? What is lim x→ ∞

x→ ∞

f ( x) ? g ( x)

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