Chapter 2: Limits
Section 2.1: Limits, Rates of Change, and Tangent Lines
• Know how to find the average rate of change (ROC) of y = f(x) over an interval
(it's
just the slope between the two points, 
and
• Know how the Average ROC is inter preted graphically
• Know what is meant by the instantaneous ROC and how to approximate it using
the Average
ROC
- If g(x) = x2 +1, estimate the instantaneous rate of change of g(x) at x = 2
using average
rates of change. (Hint: Use two points near 2, for example 2 and 2.01.).
• Know what the tangent line is and how to compute its slope
• Know that the velocity of an object is the ROC of the position of the object
- Suppose a particle's distance from the origin is given by
. Find the average velocity
of the particle between t = 0 and t = 2 seconds. What is the instantaneous
velocity at
t = 2 seconds?
Section 2.2: Limits: A Numerical and Graphical Approach
• Know the definition of a limit. That is, know how to interpret
.
• Know when a limit does not exist
- Note: A limit may exist even if f(c) does not exist
• Know what is meant by a one sided limit and know the difference between
and
• Note: One-sided limits always exist, but the limit only exists if both
one-sided limits exist
and are equal
- Show that 
does not exist.
• Given a graph of a function , know how to determine a limit
- Below is a graph of a function f(x):
(i) Do the 
and
exist? If so, find the limits, otherwise, explain
carefully why they do not exist.
(ii) Is f(x) continuous from the left at x = 2? Why or why not?
(iii) Is f(x) continuous from the right at x = 2? Why or why not?
(iv) List and classify (e.g. "removeable", "jump", or "infinte") all points of
discontinuity
of f(x) on the interval [−1, 5].
Section 2.3: Basic Limit Laws
• Know the four basic limit laws.
• Note: The above laws only hold if
and 
exist.
Section 2.4: Limits and Continuity
• Know what it means for a function f(x) to be continuous at a point x = c.
- Let a > 0 and consider the function 
Find the value of a for which f is continuous and explain why the value of a
that you
found makes f continuous.
• Know what it means for a function f(x) just be called
continuous.
• Know what it means for a function f(x) to be right-continuous
(left-continuous) at x = c
• Know the three common types of discontinuities
1. Removable discontinuity
exists but does not
equal f(c))
- The function 
has a removable discontinuity
at x = −1
2. Jump discontinuity (the one-sided limits both exist but are not equal)
- The function 
has a jump
discontinuity at x = 2.
3. infinite discontinuity (the limit is infinite as x approaches c from one or
both sides)
- The function f(x) = 1/x has an infinite discontinuity at x = 0
• Know what the Laws of Continuity say about sums, products, multiplies, and
composites of
continuous functions
• Polynomials are continuous everywhere and rational functions are continuous
everywhere the
denominator does not equal 0.
• y = sin x and y = cos x are continuous everywhere
• For b > 0, y = bx is continuous everywhere
• For b > 0 and b ≠ 1,
is continuous for x > 0
• If n is a positive, whole number , then
is continuous everywhere if n
is odd and
continuous for x > 0 if n is even
• Know the substitution method for evaluating limits if a function f(x) is known
to be contin-
uoust at a point
Section 2.5: Evaluating Limits Algebraically
• Know what it means for a function f(x) to have an indeterminate form at a
point x = c.
• Know how to use algebraic manipulations to evaluate an indeterminate form
- Evaluate 
algebraically.
• For rational functions, try factoring the numerator and
denominator
- Evaluate 
algebraically.
• For trigonometric functions , try expressing the functions in terms of sine and
cosine
- Evaluate
).
• For functions involving square roots, try multiplying by the conjugate
- Evaluate 
algebraically.
- For each of the following limits, either evaluate the limit or explain why it
does not exist.
and 
- Evaluate 
. Be sure to show all of your
work.
Section 2.6: Trigonometric Limits
• Know what the Squeeze Theorem says and how to use it to evaluate some limits
involving
trigonometric functions.
• Two important trigonometric limits are: 
and 
• Know how changing a variable can allow one to apply the Squeeze Theorem
- Evaluate 
and
. if it exists.
Section 2.7: Intermediate Value Theorem
• Know what the Intermediate Value Theorem (IVT) says (a continuous function
takes all
values between f(a) and f(b))
• Know how to use the IVT to show that a function f(x) has a zero in (a- b) if
f(x) is continuous
on the interval [a- b].
- Let f(x) = x4 − 2x3 − x2 − 1. Show that f(x) has at least one negative root .
• Know the Bisection Method and how it can be used to
approximate the zeros of a function.
- Use the function f(x) = x2 − 2 and the starting interval [1, 2] to approximate
a root of
f (x). (Note: The Bisection Method will give you a good approximation to
.).
