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1.1 Limits: An Intuitive Approach
SUGGESTED REFERENCE MATERIAL:
As you work through the problems listed below, you should reference Chapter 1.1 of the rec- ommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes.
EXPECTED SKILLS:
- Given the graph of a function y = f (x), be able to determine the limit of f (x) as x approaches some finite value (as both a one-sided and two-sided limit).
- Know how to determine when such a limit does not exist, and if appropriate, indicate whether the behavior of the function increases or decreases without bound.
PRACTICE PROBLEMS:
Questions 1-5 refer to the function F (x), which is illustrated below.
- Compute each of the following quantities. If a limit does not exist, write +∞, −∞, or DNE (whichever is most appropriate). (a) (^) xlim→ 1 − F (x) (b) (^) xlim→ 1 + F (x) (c) lim x→ 1 F (x) (d) F (1)
- Compute each of the following quantities. If a limit does not exist, write +∞, −∞, or DNE (whichever is most appropriate). (a) (^) xlim→ 3 − F (x) (b) (^) xlim→ 3 + F (x) (c) lim x→ 3 F (x) (d) F (3)
- Compute each of the following quantities. If a limit does not exist, write +∞, −∞, or DNE (whichever is most appropriate). (a) (^) xlim→ 0 − F (x) (b) (^) xlim→ 0 + F (x) (c) lim x→ 0 F (x) (d) F (0)
- Compute each of the following quantities. If a limit does not exist, write +∞, −∞, or DNE (whichever is most appropriate). (a) (^) x→−lim 1 − F (x) (b) (^) x→−lim 1 + F (x) (c) (^) xlim→− 1 F (x) (d) F (−1)
- Compute each of the following quantities. If a limit does not exist, write +∞, −∞, or DNE (whichever is most appropriate). (a) (^) x→−lim 3 − F (x) (b) (^) x→−lim 3 + F (x) (c) (^) xlim→− 3 F (x) (d) F (−3)
- Compute each of the following quantities. If a limit does not exist, write +∞, −∞, or DNE (whichever is most appropriate). (a) (^) x→lim 10 − G(x) (b) (^) x→lim 10 + G(x) (c) (^) xlim→ 10 G(x) (d) G(10)
Questions 10-12 refer to the graph of H(x), which is illustrated below.
- Compute each of the following quantities. If a limit does not exist, write +∞, −∞, or DNE (whichever is most appropriate). (a) (^) x→−lim 2 − H(x) (b) (^) x→−lim 2 + H(x) (c) (^) xlim→− 2 H(x) (d) H(−2)
- Compute each of the following quantities. If a limit does not exist, write +∞, −∞, or DNE (whichever is most appropriate). (a) (^) xlim→ 0 − H(x) (b) (^) xlim→ 0 + H(x)
(c) lim x→ 0 H(x) (d) H(0)
- Compute each of the following quantities. If a limit does not exist, write +∞, −∞, or DNE (whichever is most appropriate). (a) (^) xlim→ 2 − H(x) (b) (^) xlim→ 2 + H(x) (c) lim x→ 2 H(x) (d) H(2)
- Let f (x) =
2 − x if x < 0 6 − x^2 if 0 < x < 3 x − 6 if x ≥ 3 Sketch the graph of f (x) and use your graph to compute each of the following: (a) (^) xlim→ 0 − f (x) (b) (^) xlim→ 0 + f (x) (c) lim x→ 0 f (x) (d) f (0) (e) (^) xlim→ 3 − f (x) (f) (^) xlim→ 3 + f (x) (g) lim x→ 3 f (x) (h) f (3)
- Sketch the graph of a function y = f (x) which satisfies the following conditions. (There are many possible answers.)
- The domain is (− 1 , 2].
- f (1) = f (2) = 5
- (^) xlim→ 1 − f (x) = 4
- (^) x→−lim 1 + f (x) = −∞
1.2 Computing Limits
SUGGESTED REFERENCE MATERIAL:
As you work through the problems listed below, you should reference Chapter 1.2 of the rec- ommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes.
EXPECTED SKILLS:
- Know the basic properties of limits; i.e., be familiar with how limits ”interact” with sums, differences, products, and other operations. See Theorem 1.2.2.
- Given the formula of a function y = f (x), be able to determine the limit of f (x) as x approaches some finite value (as both a one-sided and two sided limit).
- Be able to determine when such a limit does not exist, and if appropiate, indicate if the behavior of the function is increasing or decreasing without bound.
- Be familiar with the indeterminate forms of^00 and ±∞±∞. And, know how to use algebraic techniques such as factoring and rationalizing to help compute these types of limits.
PRACTICE PROBLEMS: In each problem, compute the limit. If the limit doesn’t exist write +∞, −∞, or DNE (whichever is most appropriate).
- Given that lim x→ 1 f (x) = 4 and lim x→ 1 g(x) = 2, determine each of the following limits: (a) lim x→ 1 (f (x) + g(x)) (b) lim x→ 1 (5f (x) − g(x)) (c) lim x→ 1
(f (x) g(x)
- lim x→ 1 (x^2 + 1)
- lim x→ 4 1
- (^) xlim→− 1 (x + 1)(x^3 )
- (^) xlim→ 5 −
(x (^2) − 6 x x^3 − 1
- (^) xlim→− 1
(x (^2) − 1 x + 1
- (^) xlim→ 2 −
(x (^2) − 4 x + 4 x − 2
- (^) xlim→ 3 +
(x (^2) + 2x − 15 x − 3
- lim x→ 1
(x (^3) − 3 x (^2) − x + 3 x^2 − 1
- (^) xlim→ 16
(√x − 4 x − 16
- lim x→ 0
(|x| x
- (^) xlim→ 4 −
( (^) x x − 4
- (^) xlim→ 4 +
( (^) x x − 4
- lim x→ 4
( (^) x x − 4
- (^) xlim→− 2
x − 2
- (^) x→−lim 2 −
( (^) x x^2 + 2x
- (^) x→−lim 2 +
( (^) x x^2 + 2x
- lim x→ 3
( (^) x 3 |x − 3 |
- (^) xlim→ 1 −
( (^) x − 1 x^2 − 2 x + 1
- (^) xlim→ 1 +
( (^) x − 1 x^2 + 2x − 3
- Let f (n) =
n^2 + 1, if n ≤ − 1 3 n + 1, if n > − 1. Compute^ nlim→−^1 f^ (n)
1.3 Limits at Infinity
SUGGESTED REFERENCE MATERIAL:
As you work through the problems listed below, you should reference Chapter 1.3 of the rec- ommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes.
EXPECTED SKILLS:
- Be able to determine limits at infinity - especially for polynomials, rational functions, functions involving radicals, exponential functions, and logarithmic functions.
- Use algebraic techniques to help with indeterminate forms of ±∞±∞ and ∞ − ∞.
- Use substitutions to evaluate limits of compositions of functions.
PRACTICE PROBLEMS:
- Based on the graph of F (x) shown below, compute the indicated limits. (Make rea- sonable assumptions about the behavior of the function outside of the shown region.)
(a) (^) x→lim+∞ F (x) (b) (^) x→−∞lim F (x)
- Based on the graph of G(x) shown below, compute the indicated limits. (Make rea- sonable assumptions about the behavior of the function outside of the shown region.)
(a) (^) x→lim+∞ G(x) (b) (^) x→−∞lim G(x)
For problems 3-32, compute the limit. If the limit doesn’t exist write +∞, −∞, or DNE (whichever is most appropriate).
- (^) xlim→∞ 1
- (^) xlim→∞^ (x^2 + 1)
- (^) xlim→∞ x^2 (x − 7)(5 − x)
- (^) xlim→∞
5 − (^) x^1 + x^43
- (^) x→−∞lim
x^200
- (^) xlim→∞
( 3 x + 2 x
- (^) xlim→∞
(−x + 3 x^2
- (^) xlim→∞
( (^) x 2 x − 1
- (^) xlim→∞
x^2 + 8x − 15 − x
- (^) xlim→∞
x + √x^2 + 2x
- (^) x→−∞lim
x + √x^2 + 2x
- (^) xlim→∞
x^2 − x − x
- A tank contains 5000 liters of pure water. Brine containing 30 grams of salt per liter of water is pumped into the tank at a rate of 25 liters per minute. It can be shown thta the concentration of salt in the tank after t minutes is: C(t) = (^) 200 +^30 t t What happens as t → ∞?
Use the following two definitions to answer questions 33-36.
- Definition: A function f (x) has a horizontal asymptote of y = L if at least one of the following is true: ◦ (^) xlim→∞ f (x) = L ◦ (^) x→−∞lim f (x) = L.
- Definition: A function f (x) has a vertical asymptote of x = a if at least one of the following is true: ◦ (^) xlim→a− f (x) = +∞ ◦ (^) xlim→a− f (x) = −∞ ◦ (^) xlim→a+ f (x) = +∞ ◦ (^) xlim→a+ f (x) = −∞
- Compute the equations of all horizontal asymptotes and vertical asymptotes, if any, for each of the following functions. (a) f (x) = (^) x 4 −x 3 (b) f (x) = x
(^2) − 5 x + 4 x^2 − 6 x + 8
- Let y = f (x) satisfy the following:
- (^) xlim→∞ f (x) = ∞
- (^) x→−∞lim f (x) = − 7
- (^) xlim→ 6 + f (x) = ∞
- (^) xlim→ 6 − f (x) = −∞ Based on this information, determine equations for the horizontal and vertical asymp- totes of f (x).
- Sketch a function y = f (x) which satisfies the following conditions. (There are many possible answers.)
- f (2) = 0
- (^) xlim→∞ f (x) = (^) x→−∞lim f (x) = 0
- (^) x→−lim 3 − f (x) = ∞
- (^) x→−lim 3 + f (x) = −∞
- (^) xlim→ 0 f (x) = −∞
- Determine whether the following statement is true or false. If the statement is true, explain why. If the statement is false, provide a specific counterexample. “A function y = f (x) can have at most two horizontal asymptotes.”
- Consider f (x) = x^2 + 1.
(a) Estimate the area between the graph of f (x) and the x-axis on the interval [0, 6] using 2 rectangles of equal width and right endpoints, as in the diagram below. Is your estimate an overestimate or an underestimate of the actual area?
1.5 Continuity
SUGGESTED REFERENCE MATERIAL:
As you work through the problems listed below, you should reference Chapter 1.5 of the rec- ommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes.
EXPECTED SKILLS:
- Know what it means for a function to be continuous at a specific value and on an interval.
- Find values where a function is not continuous; specifically, you should be able to do this for polynomials, rational functions, exponential and logarithmic functions, and other elementary functions.
- Determine the values for which a piecewise function is discontinuous, if any such values exist.
- Use the Intermediate Value Theorem to show the existence of a solution to an equation.
PRACTICE PROBLEMS: Use the graph of f (x), shown below, to answer questions 1-
- For which values of x is f (x) discontinuous?
- At each point of discontinuity, explain why f (x) is discontinuous.
- Determine whether f (x) is continuous on the given interval. If not, explain why. (a) [− 8 , −4] (b) [− 8 , 0] (c) [− 8 , 0) (d) [− 2 , 1] (e) (3, 6) (f) [3, 6) (g) (6, 9] (h) [6, 9]
- For each of the following, sketch the graph of a function, y = f (x), which satisfies the given characteristic. (There are many possible answers for each) (a) f (x) is continuous everywhere except at x = 1. (b) f (x) is continuous everywhere except at x = −2 where the (^) x→−lim 2 − f (x) = (^) x→−lim 2 + f (x). (c) f (x) is continuous everywhere except at x = 0, where f (0) = 2.
- Sketch the graph of a function which satisfies the following criteria:
- The domain of f (x) is [1, 3]
- f (x) is continuous on [1, 2] and (2, 3].
- f (x) is not continuous on [1, 3]
For problems 6-15, determine the value(s) of x where the given function has a point of discontinuity, if any such values exist.
- f (x) = |x|
- f (x) = x^2 − x − 5
- f (x) = (^) x −x 1
- f (x) = √^3 x − 1
- f (x) = x (^2) + 3x − 10 x − 7
- f (x) = x
x − 2
For each of the follwing, determine the value(s) of x where the given function has a point of discontinuity. Classify each discontinuity as a removable discontinuity, a jump discontinuity, or neither. (a) f (x) = x
x − 2 (b) f (x) = x x^ −−^14 (c) f (x) =
x^2 − 3 x + 4, if x ≤ 1 x^4 − 4 x^3 − 2 x^2 + 6, if x > 1 (d) f (x) = (^) x (^2) −x^ − 4 x^1 + 3
- Multiple Choice: Consider the function:
f (x) =
x^2 if x < − 2 4 if − 2 < x ≤ 1 6 − x if x > 1 Which of the following statements is true about f (x)? (a) f (x) is continuous everywhere. (b) If f (−2) were defined to be 4, then f (x) would be continuous everywhere. (c) The only discontinuity of f (x) occurs when x = −2. (d) The only discontinuity of f (x) occurs when x = 1. (e) The only discontinuities of f (x) occur when x = −2 and x = 1.
- Show that the equation x^3 − x^2 + 3x − 1 = 1 has at least one solution in (0, 1).
- Show that f (x) = x^3 − 9 x + 5 has at least one x-intercept in (1, 10).
- Use the intermediate value theorem to show that x^3 − 2 x^2 − 2 x + 1 = 0 has at least TWO solutions in [0, 5].
1.6 Limits & Continuity of Trigonometric Functions
SUGGESTED REFERENCE MATERIAL:
As you work through the problems listed below, you should reference Chapter 1.6 of the rec- ommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes. EXPECTED SKILLS:
- Know where the trigonometric and inverse trigonometric functions are continuous.
- Be able to use lim x→ 0 sinx^ x= 1 or lim x→ 01 −^ xcos x= 0 to help find the limits of functions involving trigonometric expressions, when appropriate.
- Understand the squeeze theorem and be able to use it to compute certain limits.
PRACTICE PROBLEMS: Evaluate the following limits. If a limit does not exist, write DNE, +∞, or −∞ (whichever is most appropriate).
- (^) xlim→ π 4 sin (2x)
- lim θ→π (θ cos θ)
- (^) xlim→ 0 + csc x
- (^) xlim→ π 2 +^
tan x
- (^) xlim→ π 2 −^ tan x
- (^) xlim→ π 4 sec x
- lim x→ 0
(sin x 3 x
- lim x→ 0
(sin 3x 3 x
- lim x→ 0
(sin x |x|
- lim x→ 0
( 1 − cos x 4 x
