MATH 160 Chapter Study Guide
Chapter 2 Limits & Continuity
2.3:
Precise
Definition
of Limit
2.2: Calculating Limits and Using the Limit Laws
2.1: Rates of Change and Tangents to Curves
Section
Concepts
Recommended
Problems
Given a function y = f(x) (either by an equation or a graph) find the average rate of change of the dependent
variable with respect to the independent variable over a given interval.
o Interpret the average rate of change physically (e.g. as average velocity or average rate of growth or decay)
and graphically (as the slope of a secant line).
o Describe the connection between average rate of change and instantaneous rate of change.
Pgs. 46 – 47; #1 – 6 & 15,
16, 17.
Pg. 99; Questions to Guide
Review #1, 2.
Explain why the idea of limits is needed.
o Give examples of problems that can’t be solved using only arithmetic and algebra but can be solved using
the idea of a limit.
o Explain why these problems can’t be solved using only arithmetic and algebra.
o Explain how these problems can be solved using limits (and arithmetic and algebra, of course).
Interpret instantaneous rate of change graphically.
o Estimate instantaneous rate of change a function y = f(x) at a point x0 from a graph of the function by
sketching the appropriate tangent line and estimating its slope.
Given an expression (equation) for a function y = f(x), find the slope of the line tangent to its graph at a specified
point x0 by making a table of slopes of secant lines across appropriately selected intervals and inferring the
slope of the tangent line from the table.
o Explain how the idea of the limit of a function is the basis for this process.
Given an expression (equation) for function y = f(x), find the instantaneous rate of change of the function at a
specified at a specific point by making a table of average rates of change over appropriately selected intervals
and inferring the instantaneous velocity from the table.
o Explain how the idea of the limit of a function is the basis for this process.
Pgs. 47 – 48; #21, 22.
Pgs. 46 – 47; #7 – 14 (with
these instructions).
Pg. 47; #18, 19, 20.
Explain in common, non-technical, language what it means to say that a function y = f(x) has limit L as x
approaches c.
Explain the difference between evaluating the limit of a function y = f(x) as x approaches c and evaluating the
function at x = c.
o Explain why the limit of a function y = f(x) as x approaches c doesn’t depend on the value f(c) of the
function at x = c. Illustrate with examples
Given a graph of a function:
(i) determine the limit of the function at specified points and
(ii) identify points where the limit does not exist.
(iii) Based on what it means to say that a function y = f(x) has limit L as x approaches c, explain why the
function has a particular limit or the limit does not exist.
Pgs. 56; #7 – 10.
Pg. 56; #1 – 4, #5-6
(create the graphs).
Pg. 59; #81(a) and 82(a).
Evaluate limits as x approaches c numerically (including the possibility that the limit might not exist).
o Explain why a table showing (exact) values of a function y = f ( x ) for selected x values close to c
cannot, in itself, conclusively establish the value of the limit of the function as x approaches c.
Pg. 58; #67 – 74.
Calculator Lab #1
Use the Limit Theorems (called “Limit Laws” in sec. 2.2), including the Sandwich Theorem, to evaluate limits
algebraically (including the possibility that the limit might not exist or the limit may be infinite). Show details of
algebra.
o Explain specifically how the Limit Theorems are used to evaluate a specific limit.
(One or more of the following algebraic techniques may be needed: factoring and dividing out common
factors, adding/subtracting rational functions, dividing rational functions and/or numerical fractions,
rationalizing numerator or denominator.)
Pgs. 56 – 57; #11 – 50.
Pg. 57; #57-62,
Pg. 57 – 58; #63 – 66
(Sandwich Theorem),
Pg. 100 – 101; #9 – 24.
Pg. 104; #25 – 30 (for
mathophiles).
Given information about the limits of several functions as x approaches some number c, use the Limit
Theorems to find the limits of combinations of these functions.
Pg. 57; #51 – 56.
Pg. 100; #3 – 5.
Pg. 103; #20
Interpret the limit concept and the formal definition of limit in “real-world” situations.
Pg. 67; #55.
Pg. 102; #4, 5, 6.
MATH 160 Study Guide; Chapter 2 Limits & Continuity
Section
Concepts
Given that limx→c f ( x ) = L where f ( x) is a specific (not necessarily linear) function and L and c are specific
2.4 One-Sided Limits
numbers, and given a specific error tolerance ε, find an associated δ graphically (using a graphing calculator).
o Show clearly how you found the associated δ.
o Explain how the ε-interval and the δ-interval are related.
Recommended
Problems
Pgs. 65 – 66; #7 – 14.
Pg. 66; in #15 – 26 find d
graphically. Calculator Lab
#2.
Give a graphical example of a function that does not have a limit at a specified point x = c.
o Explain how to see from your graph that no matter what number L is chosen, the function in your example
is not related to the number L in the way the definition of limx→c f ( x ) = L requires.
Pgs. 67 – 68; #57, 58, 59,
60.
Given a statement commonly used to (incorrectly) describe what limx→c f ( x ) = L means, give examples that
show the statement is wrong.
o Explain how your example shows the statement is wrong
Pg. 67; #53, 54.
(i) Explain in common, non-technical, language what it means to say that a function y = f(x) has right-hand limit
L as x approaches c.
(ii) Explain in common, non-technical, language what it means to say that a function y = f(x) has left-hand limit
L as x approaches c.
(iii) Use correct notation to indicate one-sided limits.
Explain the difference between and the connection between one-sided limits and (two-sided) limits.
o Explain why one would expect one-sided limits and (two-sided) limits to be connected in this way.
Evaluate one-sided limits graphically and algebraically.
o Use information about the one-sided limits to infer the existence or non-existence of a limit.
Pgs. 73 – 75; #1 – 20.
Pg. 75; #43, 44.
State the mathematical definition of the phrase “a function y = f ( x ) is continuous at x = c.”
Identify points where a function is continuous and where it is not continuous from a graph of the function.
Explain in terms of the definition why the function is or is not continuous at the point(s) you identified.
Pg. 84; #1 – 10.
2.6 Limits
Involving
Infinity;
Asymptotes
of Graphs
2.5 Continuity
Give graphical examples of functions that:
(i) are not continuous at a point c but do have a limit as x approaches c;
(ii) are not continuous at a point c but are defined at the point x = c;
(iii) are not continuous at a point c, but have both a left-hand and a right-hand limit as x approaches c.
NOTE: In each case, if it is not possible to give such an example, explain why.
Use the facts (theorems) that algebraic combinations and compositions of continuous functions are continuous
to explain how to see that a given function is continuous.
o Then use the fact that the function is continuous to justify evaluating its limit by substitution.
Pg. 84; #11 – 28.
Given a function (that may be defined piecewise) and a specific point x = c, use the definition to determine
whether the function (i) is continuous, or (ii) has a removable discontinuity , or (iii) has a jump discontinuity, (iv)
has an oscillating discontinuity, or (v) has an infinite discontinuity at the point x = c .
o Explain how your conclusion follows from the definitions.
o If the function has a removable discontinuity at the point x = c, determine what value should be assigned
to f(c) so as to make the function be continuous at the point x = c.
Pg. 85; #39 – 42.
Given a function defined (piecewise) by expressions that involve unknown constants, use the definition to
determine the constants so that the function is continuous at specified points or to show that there are no values
for the constants that make the function continuous at points in question.
o Explain how your conclusion follows from the definition of continuity of a function at a point.
Pg. 84; #29, #30.
Pg. 85; #43 - 48.
Write a complete statement of the Intermediate Value Theorem.
o Explain how to see graphically that any continuous function whose domain is an interval of real numbers
must have the Intermediate Value Property. (In other words, explain the Intermediate Value Theorem
graphically.)
o Use the Intermediate Value Theorem to explain why a given function must take on certain values and why
certain equations must have solutions.
Pg. 85; #53 – 57.
Pg. 86; #65, 66, 67,
Pg. 101; #29, 30.
Pg. 103; #19.
Explain common, non-technical, language what it means for a function to have a limit L as x approaches plus
infinity or minus infinity. Use a graph to illustrate your explanation.
o _ Explain what is meant by a horizontal asymptote of a graph.
o _ Explain how limits as x approaches ± ∞ and horizontal asymptotes are related.
Pg. 97; #1 – 2.
Pg. 98; #77, #79.
Explain why you would expect the familiar Limit Theorems for sums, difference, products, and quotients of
functions to be true for limits as x approaches plus infinity or minus infinity. (Compare Th. 12, pg. 85.)
MATH 160 Study Guide; Chapter 2 Limits & Continuity
Section
Concepts
Evaluate limits as x approaches ± ∞ numerically and graphically (including the possibility that the limit might
not exist).
o Explain why a table showing (exact) values of a function y = f ( x ) for selected large x values cannot, in
itself, conclusively establish the value of the limit of the function as x approaches ± ∞.
Evaluate limits as x approaches infinity or minus infinity algebraically. Show the details of the algebraic work.
Use a calculator to confirm the results numerically and/or graphically.
o Explain how to see that the limit as x approaches ±∞ of the ratio of two polynomials of the same degree
is the ratio of their leading coefficients (see Example 3, Pg. 86).
Determine the horizontal asymptotes of the graph of a function by evaluating the limit of the function as x
approaches ±∞
Recommended
Problems
Pg. 97; #3 – 8 (Change
instructions to fit this
statement.).
Pg. 97; #9 – 36.
Pgs. 98 – 99; #80 – 85.
Pg. 101; 37 – 46.
Pg. 101; #48.
Explain in common, non-technical, language what it means when we say a function has infinity or minus infinity
as it limit as x approaches a number a and as x approaches a number a from the left or from the right.
o Explain the difference between having a limit and having an infinite limit. In particular, explain why it is that
a function can at the same time have an infinite limit and not have a limit.
Explain what is meant by a vertical asymptote.
o Explain how infinite limits and vertical asymptotes are related. Use graphs to illustrate your explanations.
Pg. 97; #1 – 2.
Pg. 98; #63 – 68
Explain the connection between one-sided limits and limits (aka two-sided limits), including infinite limits.
o Use information about the one-sided limits to infer the existence or non-existence of a limit (including
infinite limits).
o Evaluate limits (including one-sided limits) algebraically, including the possibility that the limit might not
exist or might be plus or minus infinity.
Pgs. 97 – 98; #37 – 62.
Sketch the graph of a function that has specified vertical and horizontal asymptotes and, perhaps, specified
values at certain points.
o Asymptotes may be given explicitly or have to be inferred from statements about limits of the function.
Pg. 98; #69 – 76.
Use algebra to locate points where the graph of a given rational function might have vertical asymptotes.
o Then evaluate the relevant limits to determine whether the graph actually has vertical asymptotes at these
points.
o Use your calculator to confirm your findings numerically and/or graphically.
Pg. 98; #63 – 68.
Pg. 101; #47 -- Create
your own problems like
this. Use polynomials of
degree no greater than 4.