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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