In this set of supplemental notes, we are going to expand on the topic of limits by providing you some rules that will make finding them easier. So let me state these rules.

FACT:

1.

SUM RULE:

2.

DIFFERENCE RULE:

3.

PRODUCT RULE:

4.

CONSTANT MULTIPLE RULE:

5.

QUOTIENT RULE:

6.

POWER RULE:

If m and n are integers, then

provided L^{m/ n} is a real number.

EXAMPLE 1:

SOLUTION: To find this limit, just plug in 2 and evaluate.

EXAMPLE 2:

SOLUTION:

Here are a couple more facts that we can us when evaluating limits.

FACT:

If P (x) = a _n x ⁿ + a _{n -1} x ^{n - 1} + . . . + a ₀, then

FACT:

If P (x) and Q (x) are polynomials and Q (c) ¹ 0, then

EXAMPLE 3:

SOLUTION:

EXAMPLE 4:

SOLUTION:

EXAMPLE 5:

SOLUTION:

If the denominator is zero at the value we are approaching, then canceling common factors out of the numerator and denominator will sometimes reduce the fraction to one whose denominator is no longer zero at that value.

EXAMPLE 6:

SOLUTION:

EXAMPLE 7:

SOLUTION:

EXAMPLE 8:

SOLUTION: To be able to cancel out a common factor, we first must rationalize the numerator to create a common factor. To rationalize the numerator multiply both top and bottom by the conjugate of the numerator.

EXAMPLE 9:

SOLUTION:

Sometimes we are given a limit that we cannot directly evaluate. But if we can "sandwich" the limit between two limits going to the same value, then the given limit goes to the same value. There is a theorem that states this fact and it is called the Sandwich (or Squeeze) Theorem.

THEOREM:

Suppose g (x) £ f (x) £ h (x) for all x in some open interval containing c, except possibly at x = c itself. Suppose also that

We will use this theorem later when we start to talk about the derivatives of the trigonometric functions.

Because of their connection with secant lines, tangents, and instantaneous rates of change, limits of the form

occur frequently in calculus. Remember that

is the slope of the secant line. As the distance between the two x-values gets smaller, the secant lines becomes the tangent line. (See figure 1) For this example, f (x) = x ².

figure 1

h = 4

y = 4x

h = 1

y = x

h = 3

y = 3x

h = 1/2

y = .5x

h = 2

y = 2x

h = 1/10

y = .1x

Notice that as h approaches zero, the slope of the secant line goes to zero, and this gives us the tangent line to f (x) at x = 0.

EXAMPLE 10:

Evaluate the limit

for f (x) = x ² at x = -2.

SOLUTION:

EXAMPLE 11:

Evaluate the limit