Save
AP Calculus BC
Unit 1: Limits and Continuity
1.6 Determining Limits Using Algebraic Manipulation
Save
Share
Learn
Content
Leaderboard
Share
Learn
Cards (38)
One technique for determining limits is to apply L'Hôpital's
Rule
What does GCF stand for when factoring polynomials?
Greatest Common Factor
The difference of squares pattern is
a
2
−
b
2
a^{2} - b^{2}
a
2
−
b
2
, which factors as (a + b)(a - b)</latex> using the difference of squares
Perfect square trinomials can be factored into the form
(
a
+
b
)
2
(a + b)^{2}
(
a
+
b
)
2
or
(
a
−
b
)
2
(a - b)^{2}
(
a
−
b
)
2
.
Give an example of factoring by grouping for the expression
2
x
3
−
3
x
2
+
2x^{3} - 3x^{2} +
2
x
3
−
3
x
2
+
4
x
−
6
4x - 6
4
x
−
6
.
(
x
2
+
(x^{2} +
(
x
2
+
2
)
(
2
x
−
3
)
2)(2x - 3)
2
)
(
2
x
−
3
)
When factoring polynomials, the first step is to extract the Greatest Common
Factor
Steps to simplify rational expressions
1️⃣ Factor both the numerator and denominator
2️⃣ Identify common factors
3️⃣ Cancel common factors
4️⃣ Simplify the resulting expression
Rationalizing numerators or denominators involves multiplying by the conjugate of the expression containing the
radical
.
The conjugate of
a
+
a +
a
+
b
\sqrt{b}
b
is
a
−
b
a - \sqrt{b}
a
−
b
, while the conjugate of a - \sqrt{b}</latex> is
a
+
a +
a
+
b
\sqrt{b}
b
.Conjugate
Rationalize the denominator of
\frac{1}{\sqrt{x} +
2}
.
x
−
2
x
−
4
\frac{\sqrt{x} - 2}{x - 4}
x
−
4
x
−
2
What is the purpose of rationalizing numerators or denominators?
Eliminate radicals
If the expression is
a
+
a +
a
+
b
\sqrt{b}
b
, the conjugate is a - \sqrt{b}</latex>
Steps to rationalize the denominator of a fraction
1️⃣ Identify the part with the radical you want to eliminate
2️⃣ Find the conjugate
3️⃣ Multiply both the numerator and denominator by the conjugate
4️⃣ Simplify the resulting expression
The conjugate of \sqrt{x} + 2</latex> is
x
−
2
\sqrt{x} - 2
x
−
2
The simplified expression after rationalizing
\frac{1}{\sqrt{x} +
2}
is \frac{\sqrt{x} - 2}{x - 4}</latex>
Match the trigonometric identity type with its corresponding formula:
Pythagorean Identities ↔️
sin
2
(
x
)
+
\sin^{2}(x) +
sin
2
(
x
)
+
cos
2
(
x
)
=
\cos^{2}(x) =
cos
2
(
x
)
=
1
1
1
Sum and Difference Identities ↔️
sin
(
a
±
b
)
=
\sin(a \pm b) =
sin
(
a
±
b
)
=
sin
(
a
)
cos
(
b
)
±
cos
(
a
)
sin
(
b
)
\sin(a)\cos(b) \pm \cos(a)\sin(b)
sin
(
a
)
cos
(
b
)
±
cos
(
a
)
sin
(
b
)
Double Angle Identities ↔️
cos
(
2
x
)
=
\cos(2x) =
cos
(
2
x
)
=
cos
2
(
x
)
−
sin
2
(
x
)
\cos^{2}(x) - \sin^{2}(x)
cos
2
(
x
)
−
sin
2
(
x
)
The expression
cos
2
(
x
)
−
sin
2
(
x
)
\cos^{2}(x) - \sin^{2}(x)
cos
2
(
x
)
−
sin
2
(
x
)
is equivalent to
cos
(
2
x
)
\cos(2x)
cos
(
2
x
)
To simplify rational expressions, the first step is to factor the numerator and
denominator
.
To factor the numerator
x
2
−
4
x^{2} - 4
x
2
−
4
, we use the difference of squares formula to get (x + 2)(x - 2)
Canceling common factors in a rational expression involves dividing both the numerator and
denominator
by the same factor.
What is the first step in simplifying rational expressions?
Factor numerator and denominator
Steps to simplify rational expressions
1️⃣ Factor the numerator and denominator
2️⃣ Identify common factors
3️⃣ Cancel common factors
4️⃣ Simplify the resulting expression
Rationalizing numerators or denominators involves multiplying by the
conjugate
of the expression containing the radical.
When multiplying by the conjugate, you are only multiplying the numerator to eliminate the radical.
False
What is the conjugate of
x
+
\sqrt{x} +
x
+
2
2
2
?
x
−
2
\sqrt{x} - 2
x
−
2
To simplify the expression after multiplying by the conjugate, you need to expand and cancel
terms
Rationalizing the denominator involves multiplying both the numerator and denominator by the
conjugate
of the denominator.
Which Pythagorean identity states that
sin
2
(
x
)
+
\sin^{2}(x) +
sin
2
(
x
)
+
cos
2
(
x
)
=
\cos^{2}(x) =
cos
2
(
x
)
=
1
1
1
?
Pythagorean Identity
Match the trigonometric identity type with its corresponding formula:
Pythagorean Identity ↔️
sin
2
(
x
)
+
\sin^{2}(x) +
sin
2
(
x
)
+
cos
2
(
x
)
=
\cos^{2}(x) =
cos
2
(
x
)
=
1
1
1
Sum and Difference Identity ↔️
sin
(
a
+
b
)
=
\sin(a + b) =
sin
(
a
+
b
)
=
sin
(
a
)
cos
(
b
)
+
\sin(a)\cos(b) +
sin
(
a
)
cos
(
b
)
+
cos
(
a
)
sin
(
b
)
\cos(a)\sin(b)
cos
(
a
)
sin
(
b
)
Double Angle Identity ↔️
cos
(
2
x
)
=
\cos(2x) =
cos
(
2
x
)
=
cos
2
(
x
)
−
sin
2
(
x
)
\cos^{2}(x) - \sin^{2}(x)
cos
2
(
x
)
−
sin
2
(
x
)
Simplifying \cos^{2}(x) - \sin^{2}(x)</latex> using trigonometric identities results in
\cos(2x)
Trigonometric identities are used to rewrite and simplify trigonometric
expressions
by applying well-known relationships.
Which double angle identity is equivalent to
cos
2
(
x
)
−
sin
2
(
x
)
\cos^{2}(x) - \sin^{2}(x)
cos
2
(
x
)
−
sin
2
(
x
)
?
cos
(
2
x
)
\cos(2x)
cos
(
2
x
)
Recognizing \cos^{2}(x) - \sin^{2}(x)</latex> as the
cos
(
2
x
)
\cos(2x)
cos
(
2
x
)
identity is the first step in simplification
What is L'Hôpital's Rule used to evaluate?
Indeterminate limits
Before applying L'Hôpital's Rule, you must verify that the limit is in an
indeterminate form
.
When differentiating the numerator and denominator of
lim
x
→
2
x
2
−
4
x
−
2
\lim_{x \to 2} \frac{x^{2} - 4}{x - 2}
lim
x
→
2
x
−
2
x
2
−
4
, you get
2
x
1
\frac{2x}{1}
1
2
x
, which simplifies to 4
What is the value of \lim_{x \to 2} \frac{2x}{1}</latex>?
4
4
4
Steps to apply L'Hôpital's Rule
1️⃣ Verify that the limit is in an indeterminate form
2️⃣ Differentiate the numerator and denominator
3️⃣ Evaluate the new limit
4️⃣ Simplify if necessary