AP CalculusChapter 1, Section 3

Evaluating Limits Analytically2013 - 2014

Some Basic Limits

• In some cases, the limit can be evaluated by direct substitution.

Evaluate the following limits

Properties of Limits

• Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits.

and

1. Scalar multiple: 2. Sum or difference: 3. Product: 4. Quotient: provided 5. Power:

lim𝑥→ 2

(4 𝑥2+3)

Limits of Polynomials & Rational Functions

• If p is a polynomial function and c is a real number, then

• If r is a rational function given by r and c is a real number such , then

lim𝑥→ 1

𝑥2+𝑥+2𝑥+1

The Limit of a Function Involving a Radical

• Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for if n is even.

The Limit of a Composite Function

• If f and g are functions such that and , then

Because

and

it follows that

Limits of Trigonometric Functions

lim𝑥→ 0

tan𝑥

lim𝑥→𝜋

(𝑥 cos 𝑥)

lim𝑥→ 0

sin2 x

Dividing Out Technique

lim𝑥→−3

𝑥2+𝑥−6𝑥+3

Rationalizing Technique

lim𝑥→ 0

√𝑥+1−1𝑥Check your answer by using a table

The Squeeze Theorem

• Basically says if you have two different function that have the same limit as , and you have a 3rd function that falls between the first two functions, the 3rd function will also have the same limit as .

for all x in an open interval containing c, except possibly c itself, and if Then exists and is equal to L.

Special Trigonometric Functions

Find the limit:

Find the limit:

Homework

• Pg. 67 – 69: #1 – 77 every other odd, 83, 87, 113