Transcript Lesson 1-1-9 powerpoint

Limits of Radical and Trig
Functions
Lesson 1.1.9
• With any limit, you can always graph
and/or make a table of values.
• However, there are more exact and less
tedious shortcuts, as we saw yesterday.
• Today, we will learn shortcuts to be used
for rational and trigonometric functions.
Learning Objectives
• Given a function with a radical binomial,
multiply the numerator and denominator
by the conjugate of that binomial to
evaluate the limit at a certain point.
• Given a trigonometric function, evaluate
the limit at a certain point.
Indeterminate Forms of Limits
• Suppose that we
were asked to
evaluate the limit on
the right.
• What would happen if
we plugged in 0 for x?
• We would end up with
the fraction 0/0.
• Of course, such a
fraction is undefined.
• When plugging in c
gives us 0/0 or
±∞/ ±∞, we say that
the limit is in
indeterminate form.
• When a limit is in this
form, we can
determine it by
manipulating the
function in some way.
• One of those ways is
by rationalizing.
Rationalizing
• In Algebra II, you
learned to rationalize
fractions with radicals
to eliminate a radical
in the denominator.
• In Calculus, you will
instead rationalize to
evaluate a limit.
3
x 2
• You can rationalize
either the numerator
or the denominator.
• You no longer have to
worry about not
leaving radicals in the
denominator
Review: What is Rationalizing?
• Rationalizing comes from
the difference of squares
concept. (a+b)(a-b) = a2 –
b2
• Notice how a+b and a-b
are the same thing, but
with the middle sign
changed. They are called
conjugates of each
other.
• Keep in mind: when you
square a square root, the
radical sign goes away:
( 3)  3
2
Therefore
• To rationalize, multiply
numerator and
denominator by the
conjugate.
3
x 2

x 2
x 2
3( x  2)

( x  2)( x  2)
3 x 6

x4
3
x 2
3 x 6

( x )2  22
Rationalizing Practice
x
x  2 1
1 x  5
2
Example 1
• Find the limit on the
right.
– First rationalize.
– Then plug in.
Trigonometric Limits
• Please know the two
limits on the right.
(Don’t worry about
why.)
Example 2
Evaluate the following limit:
Trig Identities
• Other trig limits may require you to apply
trigonometric identities. Know the ones below.
Reteaching #1
• Evaluate the following limit
Example 3
Graphs of Trig Functions
• For some trig limits, it helps to refer to the
graph of one of the basic trig functions.
• We went over sine and cosine in Lesson
1.1.3.
• Now let’s go over tan, cot, sec, csc.
y = tan x
• Note: the vertical lines are asymptotes. They are
not part of the graph.
y = cot x
y = sec x
y = csc x
Example 4
cot x
lim
x  0 cos x
• Use trig identities to
simplify.
• Use one of your
graphs to determine.
Homework
• Textbook 1a,c; 2a; 3a,c,d