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Precalculus

12.2 Finding Limits: Properties of Limits

Precalculus12.2 Finding Limits: Properties of Limits
  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    6. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Equations
    8. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  14. A | Basic Functions and Identities
  15. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  16. Index

Learning Objectives

In this section, you will:

  • Find the limit of a sum, a difference, and a product.
  • Find the limit of a polynomial.
  • Find the limit of a power or a root.
  • Find the limit of a quotient.

Consider the rational function

f(x)= x 2 6x7 x7 f(x)= x 2 6x7 x7

The function can be factored as follows:

f(x)= ( x7 ) ( x+1 ) x7 , which gives us f(x)=x+1,x7. f(x)= ( x7 ) ( x+1 ) x7 , which gives us f(x)=x+1,x7.

Does this mean the function f f is the same as the function g(x)=x+1? g(x)=x+1?

The answer is no. Function f f does not have x=7 x=7 in its domain, but g g does. Graphically, we observe there is a hole in the graph of f( x ) f( x ) at x=7, x=7, as shown in Figure 1 and no such hole in the graph of g( x ), g( x ), as shown in Figure 2.

Graph of an increasing function where f(x) = (x^2-6x-7)\(x-7) with a discontinuity at (7, 8)
Figure 1 The graph of function f f contains a break at x=7 x=7 and is therefore not continuous at x=7. x=7.
Graph of an increasing function where g(x) = x+1
Figure 2 The graph of function g g is continuous.

So, do these two different functions also have different limits as x x approaches 7?

Not necessarily. Remember, in determining a limit of a function as x x approaches a, a, what matters is whether the output approaches a real number as we get close to x=a. x=a. The existence of a limit does not depend on what happens when x x equals a. a.

Look again at Figure 1 and Figure 2. Notice that in both graphs, as x x approaches 7, the output values approach 8. This means

lim x7 f(x)= lim x7 g(x). lim x7 f(x)= lim x7 g(x).

Remember that when determining a limit, the concern is what occurs near x=a, x=a, not at x=a. x=a. In this section, we will use a variety of methods, such as rewriting functions by factoring, to evaluate the limit. These methods will give us formal verification for what we formerly accomplished by intuition.

Finding the Limit of a Sum, a Difference, and a Product

Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the properties of limits, which is a collection of theorems for finding limits.

Knowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.

Properties of Limits

Let a,k,A, a,k,A, and B B represent real numbers, and f f and g g be functions, such that lim xa f(x)=A lim xa f(x)=A and lim xa g(x)=B. lim xa g(x)=B. For limits that exist and are finite, the properties of limits are summarized in Table 1

Constant, k lim xa k=k lim xa k=k
Constant times a function lim xa [ kf(x) ]=k lim xa f(x)=kA lim xa [ kf(x) ]=k lim xa f(x)=kA
Sum of functions lim xa [ f(x)+g(x) ]= lim xa f(x)+ lim xa g(x)=A+B lim xa [ f(x)+g(x) ]= lim xa f(x)+ lim xa g(x)=A+B
Difference of functions lim xa [ f(x)g(x) ]= lim xa f(x) lim xa g(x)=AB lim xa [ f(x)g(x) ]= lim xa f(x) lim xa g(x)=AB
Product of functions lim xa [ f(x)g(x) ]= lim xa f(x) lim xa g(x)=AB lim xa [ f(x)g(x) ]= lim xa f(x) lim xa g(x)=AB
Quotient of functions lim xa f(x) g(x) = lim xa f(x) lim xa g(x) = A B ,B0 lim xa f(x) g(x) = lim xa f(x) lim xa g(x) = A B ,B0
Function raised to an exponent lim xa [f(x)] n = [ lim xa f(x) ] n = A n , lim xa [f(x)] n = [ lim xa f(x) ] n = A n , where n n is a positive integer
nth root of a function, where n is a positive integer lim xa f(x) n = lim xa [ f(x) ] n = A n lim xa f(x) n = lim xa [ f(x) ] n = A n
Polynomial function lim xa p(x)=p(a) lim xa p(x)=p(a)
Table 1

Example 1

Evaluating the Limit of a Function Algebraically

Evaluate lim x3 ( 2x+5 ). lim x3 ( 2x+5 ).

Try It #1

Evaluate the following limit: lim x12 ( 2x+2 ). lim x12 ( 2x+2 ).

Finding the Limit of a Polynomial

Not all functions or their limits involve simple addition, subtraction, or multiplication. Some may include polynomials. Recall that a polynomial is an expression consisting of the sum of two or more terms, each of which consists of a constant and a variable raised to a nonnegative integral power. To find the limit of a polynomial function, we can find the limits of the individual terms of the function, and then add them together. Also, the limit of a polynomial function as x x approaches a a is equivalent to simply evaluating the function for a a.

Given a function containing a polynomial, find its limit.

  1. Use the properties of limits to break up the polynomial into individual terms.
  2. Find the limits of the individual terms.
  3. Add the limits together.
  4. Alternatively, evaluate the function for a a.

Example 2

Evaluating the Limit of a Function Algebraically

Evaluate lim x3 ( 5 x 2 ). lim x3 ( 5 x 2 ).

Try It #2

Evaluate lim x4 ( x 3 5). lim x4 ( x 3 5).

Example 3

Evaluating the Limit of a Polynomial Algebraically

Evaluate lim x5 ( 2 x 3 3x+1 ). lim x5 ( 2 x 3 3x+1 ).

Try It #3

Evaluate the following limit: lim x1 ( x 4 4 x 3 +5 ). lim x1 ( x 4 4 x 3 +5 ).

Finding the Limit of a Power or a Root

When a limit includes a power or a root, we need another property to help us evaluate it. The square of the limit of a function equals the limit of the square of the function; the same goes for higher powers. Likewise, the square root of the limit of a function equals the limit of the square root of the function; the same holds true for higher roots.

Example 4

Evaluating a Limit of a Power

Evaluate lim x2 ( 3x+1 ) 5 . lim x2 ( 3x+1 ) 5 .

Try It #4

Evaluate the following limit: lim x4 ( 10x+36 ) 3 . lim x4 ( 10x+36 ) 3 .

Q&A

If we can’t directly apply the properties of a limit, for example in lim x2 ( x 2 +6x+8 x2 ) lim x2 ( x 2 +6x+8 x2 ), can we still determine the limit of the function as x x approaches a a?

Yes. Some functions may be algebraically rearranged so that one can evaluate the limit of a simplified equivalent form of the function.

Finding the Limit of a Quotient

Finding the limit of a function expressed as a quotient can be more complicated. We often need to rewrite the function algebraically before applying the properties of a limit. If the denominator evaluates to 0 when we apply the properties of a limit directly, we must rewrite the quotient in a different form. One approach is to write the quotient in factored form and simplify.

Given the limit of a function in quotient form, use factoring to evaluate it.

  1. Factor the numerator and denominator completely.
  2. Simplify by dividing any factors common to the numerator and denominator.
  3. Evaluate the resulting limit, remembering to use the correct domain.

Example 5

Evaluating the Limit of a Quotient by Factoring

Evaluate lim x2 ( x 2 6x+8 x2 ). lim x2 ( x 2 6x+8 x2 ).

Analysis

When the limit of a rational function cannot be evaluated directly, factored forms of the numerator and denominator may simplify to a result that can be evaluated.

Notice, the function

f(x)= x 2 6x+8 x2 f(x)= x 2 6x+8 x2

is equivalent to the function

f(x)=x4,x2. f(x)=x4,x2.

Notice that the limit exists even though the function is not defined at x = 2. x = 2.

Try It #5

Evaluate the following limit: lim x7 ( x 2 11x+28 7x ). lim x7 ( x 2 11x+28 7x ).

Example 6

Evaluating the Limit of a Quotient by Finding the LCD

Evaluate lim x5 ( 1 x 1 5 x5 ). lim x5 ( 1 x 1 5 x5 ).

Analysis

When determining the limit of a rational function that has terms added or subtracted in either the numerator or denominator, the first step is to find the common denominator of the added or subtracted terms; then, convert both terms to have that denominator, or simplify the rational function by multiplying numerator and denominator by the least common denominator. Then check to see if the resulting numerator and denominator have any common factors.

Try It #6

Evaluate lim x5 ( 1 5 + 1 x 10+2x ). lim x5 ( 1 5 + 1 x 10+2x ).

Given a limit of a function containing a root, use a conjugate to evaluate.

  1. If the quotient as given is not in indeterminate ( 0 0 ) ( 0 0 ) form, evaluate directly.
  2. Otherwise, rewrite the sum (or difference) of two quotients as a single quotient, using the least common denominator (LCD).
  3. If the numerator includes a root, rationalize the numerator; multiply the numerator and denominator by the conjugate of the numerator. Recall that a± b a± b are conjugates.
  4. Simplify.
  5. Evaluate the resulting limit.

Example 7

Evaluating a Limit Containing a Root Using a Conjugate

Evaluate lim x0 ( 25x 5 x ). lim x0 ( 25x 5 x ).

Analysis

When determining a limit of a function with a root as one of two terms where we cannot evaluate directly, think about multiplying the numerator and denominator by the conjugate of the terms.

Try It #7

Evaluate the following limit: lim h0 ( 16h 4 h ). lim h0 ( 16h 4 h ).

Example 8

Evaluating the Limit of a Quotient of a Function by Factoring

Evaluate lim x4 ( 4x x 2 ). lim x4 ( 4x x 2 ).

Analysis

Multiplying by a conjugate would expand the numerator; look instead for factors in the numerator. Four is a perfect square so that the numerator is in the form

a 2 b 2 a 2 b 2

and may be factored as

( a+b )( ab ). ( a+b )( ab ).
Try It #8

Evaluate the following limit: lim x3 ( x3 x 3 ). lim x3 ( x3 x 3 ).

Given a quotient with absolute values, evaluate its limit.

  1. Try factoring or finding the LCD.
  2. If the limit cannot be found, choose several values close to and on either side of the input where the function is undefined.
  3. Use the numeric evidence to estimate the limits on both sides.

Example 9

Evaluating the Limit of a Quotient with Absolute Values

Evaluate lim x7 | x7 | x7 . lim x7 | x7 | x7 .

Try It #9

Evaluate lim x 6 + 6x | x6 | . lim x 6 + 6x | x6 | .

Access the following online resource for additional instruction and practice with properties of limits.

12.2 Section Exercises

Verbal

1.

Give an example of a type of function f f whose limit, as x x approaches a, a, is f( a ). f( a ).

2.

When direct substitution is used to evaluate the limit of a rational function as x x approaches a a and the result is f( a )= 0 0 , f( a )= 0 0 , does this mean that the limit of f f does not exist?

3.

What does it mean to say the limit of f( x ) , f( x ) , as x x approaches c ,c, is undefined?

Algebraic

For the following exercises, evaluate the limits algebraically.

4.

lim x0 ( 3 ) lim x0 ( 3 )

5.

lim x2 ( 5x x 2 1 ) lim x2 ( 5x x 2 1 )

6.

lim x2 ( x 2 5x+6 x+2 ) lim x2 ( x 2 5x+6 x+2 )

7.

lim x3 ( x 2 9 x3 ) lim x3 ( x 2 9 x3 )

8.

lim x1 ( x 2 2x3 x+1 ) lim x1 ( x 2 2x3 x+1 )

9.

lim x 3 2 ( 6 x 2 17x+12 2x3 ) lim x 3 2 ( 6 x 2 17x+12 2x3 )

10.

lim x 7 2 ( 8 x 2 +18x35 2x+7 ) lim x 7 2 ( 8 x 2 +18x35 2x+7 )

11.

lim x3 ( x 2 9 x5x+6 ) lim x3 ( x 2 9 x5x+6 )

12.

lim x3 ( 7 x 4 21 x 3 12 x 4 +108 x 2 ) lim x3 ( 7 x 4 21 x 3 12 x 4 +108 x 2 )

13.

lim x3 ( x 2 +2x3 x3 ) lim x3 ( x 2 +2x3 x3 )

14.

lim h0 ( ( 3+h ) 3 27 h ) lim h0 ( ( 3+h ) 3 27 h )

15.

lim h0 ( ( 2h ) 3 8 h ) lim h0 ( ( 2h ) 3 8 h )

16.

lim h0 ( ( h+3 ) 2 9 h ) lim h0 ( ( h+3 ) 2 9 h )

17.

lim h0 ( 5h 5 h ) lim h0 ( 5h 5 h )

18.

lim x0 ( 3x 3 x ) lim x0 ( 3x 3 x )

19.

lim x9 ( x 2 81 3 x ) lim x9 ( x 2 81 3 x )

20.

lim x1 ( x x 2 1 x ) lim x1 ( x x 2 1 x )

21.

lim x0 ( x 1+2x 1 ) lim x0 ( x 1+2x 1 )

22.

lim x 1 2 ( x 2 1 4 2x1 ) lim x 1 2 ( x 2 1 4 2x1 )

23.

lim x4 ( x 3 64 x 2 16 ) lim x4 ( x 3 64 x 2 16 )

24.

lim x 2 ( |x2| x2 ) lim x 2 ( |x2| x2 )

25.

lim x 2 + ( | x2 | x2 ) lim x 2 + ( | x2 | x2 )

26.

lim x2 ( | x2 | x2 ) lim x2 ( | x2 | x2 )

27.

lim x 4 ( | x4 | 4x ) lim x 4 ( | x4 | 4x )

28.

lim x 4 + ( | x4 | 4x ) lim x 4 + ( | x4 | 4x )

29.

lim x4 ( | x4 | 4x ) lim x4 ( | x4 | 4x )

30.

lim x2 ( 8+6x x 2 x2 ) lim x2 ( 8+6x x 2 x2 )

For the following exercise, use the given information to evaluate the limits: lim xc f(x)=3, lim xc f(x)=3, lim xc g( x )=5 lim xc g( x )=5 .

31.

lim xc [ 2f(x)+ g(x) ] lim xc [ 2f(x)+ g(x) ]

32.

lim xc [ 3f(x)+ g(x) ] lim xc [ 3f(x)+ g(x) ]

33.

lim xc f(x) g(x) lim xc f(x) g(x)

For the following exercises, evaluate the following limits.

34.

lim x2 cos( πx ) lim x2 cos( πx )

35.

lim x2 sin( πx ) lim x2 sin( πx )

36.

lim x2 sin( π x ) lim x2 sin( π x )

37.

f(x)={ 2 x 2 +2x+1, x0 x3, x>0 lim x 0 + f(x) f(x)={ 2 x 2 +2x+1, x0 x3, x>0 lim x 0 + f(x)

38.

f(x)={ 2 x 2 +2x+1, x0 x3, x>0 lim x 0 f(x) f(x)={ 2 x 2 +2x+1, x0 x3, x>0 lim x 0 f(x)

39.

f(x)={ 2 x 2 +2x+1, x0 x3, x>0 lim x0 f(x) f(x)={ 2 x 2 +2x+1, x0 x3, x>0 lim x0 f(x)

40.

lim x4 x+5 3 x4 lim x4 x+5 3 x4

41.

lim x 2 + (2x〚x〛) lim x 2 + (2x〚x〛)

42.

lim x2 x+7 3 x 2 x2 lim x2 x+7 3 x 2 x2

43.

lim x 3 + x 2 x 2 9 lim x 3 + x 2 x 2 9

For the following exercises, find the average rate of change f(x+h)f(x) h . f(x+h)f(x) h .

44.

f(x)=x+1 f(x)=x+1

45.

f(x)=2 x 2 1 f(x)=2 x 2 1

46.

f(x)= x 2 +3x+4 f(x)= x 2 +3x+4

47.

f(x)= x 2 +4x100 f(x)= x 2 +4x100

48.

f(x)=3 x 2 +1 f(x)=3 x 2 +1

49.

f(x)=cos(x) f(x)=cos(x)

50.

f(x)=2 x 3 4x f(x)=2 x 3 4x

51.

f(x)= 1 x f(x)= 1 x

52.

f(x)= 1 x 2 f(x)= 1 x 2

53.

f(x)= x f(x)= x

Graphical

54.

Find an equation that could be represented by Figure 3.

Graph of increasing function with a removable discontinuity at (2, 3).
Figure 3
55.

Find an equation that could be represented by Figure 4.

Graph of increasing function with a removable discontinuity at (-3, -1).
Figure 4

For the following exercises, refer to Figure 5.

Graph of increasing function from zero to positive infinity.
Figure 5
56.

What is the right-hand limit of the function as x x approaches 0?

57.

What is the left-hand limit of the function as x x approaches 0?

Real-World Applications

58.

The position function s(t)=16 t 2 +144t s(t)=16 t 2 +144t gives the position of a projectile as a function of time. Find the average velocity (average rate of change) on the interval [ 1,2 ] [ 1,2 ] .

59.

The height of a projectile is given by s(t)=64 t 2 +192t s(t)=64 t 2 +192t Find the average rate of change of the height from t=1 t=1 second to t=1.5 t=1.5 seconds.

60.

The amount of money in an account after t t years compounded continuously at 4.25% interest is given by the formula A= A 0 e 0.0425t , A= A 0 e 0.0425t , where A 0 A 0 is the initial amount invested. Find the average rate of change of the balance of the account from t=1 t=1 year to t=2 t=2 years if the initial amount invested is $1,000.00.

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