AP Calculus AB: 2.1.5 The Limit Laws, Part II

Question 1

The Limit Laws, Part II

Accepted Answer

Use the definition of the limit of a function at a point to find such limits for simple functions such as f(x) = c, f(x) = x, f(x) = x^n, and polynomial and rational functions.
Understand and apply the Limit Law for Polynomial Functions, the Limit Law for Rational Functions, and the Root Law.
Understand and apply limit laws for roots and for powers of functions.

Question 2

note

Accepted Answer

The limit of a function f at a point x= a is a number L such that as x gets close to a, f(x) gets close to L. For constant functions of the form f(x) = c, as x gets close to a, f(x) is always close to (in fact equal to) c, so the limit of f(x) = c as x approaches a is c.
Thus, as shown in the example at left, the limit of 3 as x approaches 2 is 3.
For the function f(x) = x, as x gets close to a, f(x) is also getting close to a. This means that the limit of f(x) = x as x approaches a is a.
Thus, as shown in the example at left, the limit of x as x approaches 2 is 2.
For functions of the form f(x) = x^n, as x gets close to a, f(x) is getting close to an. This means that the limit of f(x) = x^n as x approaches a is an.
Thus, as the example shows, the limit of x5 as x approaches 2 is 25.
For a general polynomial function, the known limit laws for sums and products may be applied, along with the limit laws just derived for some simple functions. Use the fact that the limit of the sum of two functions is the sum of the limits of those two functions (provided the limits exist) for the sums in the polynomial. Also use the fact that the limit of the product of two functions is the product of the limits of those two functions (provided the limits exist) for the product of a constant coefficient with x raised to a power that appears in the terms of the polynomial.
Thus, as shown in the example at left, the limit as x approaches 3 of 2x2 − 4x + 7 is 2(3)2 − 4(3) + 7, or 13.
The general Limit Law for Polynomial Functions states that the limit of a polynomial function at a point is the function evaluated at that point.

Question 3

note 2

Accepted Answer

Rational functions behave similarly. The limit of a rational function at a point is equal to the function evaluated at that point (as long as the quotient is not zero at the point).
This is called the Limit Law for Rational Functions.
For the function given by the nth root of x when xis close to a, the nth root of xis close to the nth root of a.
Thus, as shown in the example at left, the limit of the cube root of x as x approaches 8 is the cube root of 8, or 2.
The limit law holds as long as when n is even, a is greater than or equal to 0. If a were negative and n were even, the limit would not exist because an even root of a negative number is undefined.
Consider a function f that can be expressed as another function g raised to a power n. As x gets close to a,f(x) gets close to the limit of g(x) as x approaches a, all raised to the power n.
If the limit as x approaches π/2 of 2sin x is known to be 2, then the limit as x approaches π/2 of (2sin x)5 is 25, or 32.
The Root Law states that the limit as x approaches a of the nth root of a function f is just the nth root of the limit of f as x approaches a, provided this limit exists, and provided that when n is even, f(x) is greater than or equal to 0 near a.
Thus, as shown in the example at left, the limit as x approaches −5 of the square root of the quantity −3x+ 1is the square root of −3(−5) + 1, or 4.

Question 4

f(x)=(x^2−4x+1)^0 Find lim x→3f(x).

Accepted Answer

1

Question 5

Which of the following statements is false?

Accepted Answer

limx→a[f(x)+g(x)]=limx→af(x)+g(x)

Question 6

f(x)=4x−3/2x+5Find limx→−2f(x).

Accepted Answer

−11

Question 7

f(x)=5√x Find limx→−32f(x).

Accepted Answer

−2

Question 8

f(x)=(2x^2−4x−8)^6Find limx→3f(x).

Accepted Answer

64

Question 9

f(x)=√x^2−14x+50 Find limx→7f(x).

Accepted Answer

1

Question 10

f(x)=x Find limx→1f(x).

Accepted Answer

1

Question 11

f(x)=6/x^2−7x+2 Find limx→7f(x).

Accepted Answer

3

Question 12

f(x)=2x^2−4x+3Find lim x→2f(x).

Accepted Answer

3

Question 13

Which of the following expressions is equal tolimx→a{c⋅f(x)g(x)−[f(x)]2}?

Accepted Answer

c⋅limx→af(x)⋅limx→ag(x)−[limx→af(x)]2

Question 14

An error has been made in evaluating this limit. Which numbered line of the evaluation of this expression represents the introduction of an error regarding the use of the limit laws?
(Assume that the limit exists and that
h (x) is never equal to zero)

Accepted Answer

Line (1)

Question 15

f(x)=√3x−7Find limx→1f(x).

Accepted Answer

None of the above

Question 16

Which numbered line of this equation represents an error in the use of the limit laws?(Assume that the limit exists)

Accepted Answer

Line (2)

Question 17

f(x)=7Find limx→−3f(x).

Accepted Answer

7

AP Calculus AB: 2.1.5 The Limit Laws, Part II

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Term	Definition
The Limit Laws, Part II	Use the definition of the limit of a function at a point to find such limits for simple functions such as f(x) = c, f(x) = x, f(x) = x^n, and polynomial and rational functions. Understand and apply the Limit Law for Polynomial Functions, the Limit Law for Rational Functions, and the Root Law. Understand and apply limit laws for roots and for powers of functions.
note	The limit of a function f at a point x= a is a number L such that as x gets close to a, f(x) gets close to L. For constant functions of the form f(x) = c, as x gets close to a, f(x) is always close to (in fact equal to) c, so the limit of f(x) = c as x approaches a is c. Thus, as shown in the example at left, the limit of 3 as x approaches 2 is 3. For the function f(x) = x, as x gets close to a, f(x) is also getting close to a. This means that the limit of f(x) = x as x approaches a is a. Thus, as shown in the example at left, the limit of x as x approaches 2 is 2. For functions of the form f(x) = x^n, as x gets close to a, f(x) is getting close to an. This means that the limit of f(x) = x^n as x approaches a is an. Thus, as the example shows, the limit of x5 as x approaches 2 is 25. For a general polynomial function, the known limit laws for sums and products may be applied, along with the limit laws just derived for some simple functions. Use the fact that the limit of the sum of two functions is the sum of the limits of those two functions (provided the limits exist) for the sums in the polynomial. Also use the fact that the limit of the product of two functions is the product of the limits of those two functions (provided the limits exist) for the product of a constant coefficient with x raised to a power that appears in the terms of the polynomial. Thus, as shown in the example at left, the limit as x approaches 3 of 2x2 − 4x + 7 is 2(3)2 − 4(3) + 7, or 13. The general Limit Law for Polynomial Functions states that the limit of a polynomial function at a point is the function evaluated at that point.
note 2	Rational functions behave similarly. The limit of a rational function at a point is equal to the function evaluated at that point (as long as the quotient is not zero at the point). This is called the Limit Law for Rational Functions. For the function given by the nth root of x when xis close to a, the nth root of xis close to the nth root of a. Thus, as shown in the example at left, the limit of the cube root of x as x approaches 8 is the cube root of 8, or 2. The limit law holds as long as when n is even, a is greater than or equal to 0. If a were negative and n were even, the limit would not exist because an even root of a negative number is undefined. Consider a function f that can be expressed as another function g raised to a power n. As x gets close to a,f(x) gets close to the limit of g(x) as x approaches a, all raised to the power n. If the limit as x approaches π/2 of 2sin x is known to be 2, then the limit as x approaches π/2 of (2sin x)5 is 25, or 32. The Root Law states that the limit as x approaches a of the nth root of a function f is just the nth root of the limit of f as x approaches a, provided this limit exists, and provided that when n is even, f(x) is greater than or equal to 0 near a. Thus, as shown in the example at left, the limit as x approaches −5 of the square root of the quantity −3x+ 1is the square root of −3(−5) + 1, or 4.
f(x)=(x^2−4x+1)^0 Find lim x→3f(x).	1
Which of the following statements is false?	limx→a[f(x)+g(x)]=limx→af(x)+g(x)
f(x)=4x−3/2x+5Find limx→−2f(x).	−11
f(x)=5√x Find limx→−32f(x).	−2
f(x)=(2x^2−4x−8)^6Find limx→3f(x).	64
f(x)=√x^2−14x+50 Find limx→7f(x).	1
f(x)=x Find limx→1f(x).	1
f(x)=6/x^2−7x+2 Find limx→7f(x).	3
f(x)=2x^2−4x+3Find lim x→2f(x).	3
Which of the following expressions is equal to limx→a{c⋅f(x)g(x)−[f(x)]2}?	c⋅limx→af(x)⋅limx→ag(x)−[limx→af(x)]2
An error has been made in evaluating this limit. Which numbered line of the evaluation of this expression represents the introduction of an error regarding the use of the limit laws? (Assume that the limit exists and that h (x) is never equal to zero)	Line (1)
f(x)=√3x−7Find limx→1f(x).	None of the above
Which numbered line of this equation represents an error in the use of the limit laws? (Assume that the limit exists)	Line (2)
f(x)=7Find limx→−3f(x).	7