2.3 The Limit Laws

In the previous section, we evaluated limits by looking at graphs or by constructing a table of values. In this section, we establish laws for calculating limits and learn how to apply these laws. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to derive the formula for the area of a circle by adapting a method devised by the Greek mathematician Archimedes. We begin by restating two useful limit results from the previous section. These two results, together with the limit laws, serve as a foundation for calculating many limits.

Evaluating Limits with the Limit Laws

The first two limit laws were stated in Two Important Limits and we repeat them here. These basic results, together with the other limit laws, allow us to evaluate limits of many algebraic functions.

Theorem 2.4

Basic Limit Results

For any real number a and any constant c,

lim x → a x = a lim x → a x = a lim x → a c = c lim x → a c = c

Example 2.13

Evaluating a Basic Limit

Evaluate each of the following limits using Basic Limit Results.

  1. lim x → 2 x lim x → 2 x
  2. lim x → 2 5 lim x → 2 5

Solution

  1. The limit of x as x approaches a is a: lim x → 2 x = 2 . lim x → 2 x = 2 .
  2. The limit of a constant is that constant: lim x → 2 5 = 5 . lim x → 2 5 = 5 .

We now take a look at the limit laws , the individual properties of limits. The proofs that these laws hold are omitted here.

Theorem 2.5

Limit Laws

Let f ( x ) f ( x ) and g ( x ) g ( x ) be defined for all x ≠ a x ≠ a over some open interval containing a. Assume that L and M are real numbers such that lim x → a f ( x ) = L lim x → a f ( x ) = L and lim x → a g ( x ) = M . lim x → a g ( x ) = M . Let c be a constant. Then, each of the following statements holds:

Sum law for limits : lim x → a ( f ( x ) + g ( x ) ) = lim x → a f ( x ) + lim x → a g ( x ) = L + M lim x → a ( f ( x ) + g ( x ) ) = lim x → a f ( x ) + lim x → a g ( x ) = L + M

Difference law for limits : lim x → a ( f ( x ) − g ( x ) ) = lim x → a f ( x ) − lim x → a g ( x ) = L − M lim x → a ( f ( x ) − g ( x ) ) = lim x → a f ( x ) − lim x → a g ( x ) = L − M

Constant multiple law for limits : lim x → a c f ( x ) = c · lim x → a f ( x ) = c L lim x → a c f ( x ) = c · lim x → a f ( x ) = c L

Product law for limits : lim x → a ( f ( x ) · g ( x ) ) = lim x → a f ( x ) · lim x → a g ( x ) = L · M lim x → a ( f ( x ) · g ( x ) ) = lim x → a f ( x ) · lim x → a g ( x ) = L · M

Quotient law for limits : lim x → a f ( x ) g ( x ) = lim x → a f ( x ) lim x → a g ( x ) = L M lim x → a f ( x ) g ( x ) = lim x → a f ( x ) lim x → a g ( x ) = L M for M ≠ 0 M ≠ 0

Power law for limits : lim x → a ( f ( x ) ) n = ( lim x → a f ( x ) ) n = L n lim x → a ( f ( x ) ) n = ( lim x → a f ( x ) ) n = L n for every positive integer n.

Root law for limits : lim x → a f ( x ) n = lim x → a f ( x ) n = L n lim x → a f ( x ) n = lim x → a f ( x ) n = L n for all L if n is odd and for L ≥ 0 L ≥ 0 if n is even and f ( x ) ≥ 0 f ( x ) ≥ 0 .

We now practice applying these limit laws to evaluate a limit.