2.3 Basic Limit Laws
In Section 2.1.4 we relied on graphical and numerical approaches to investigate limits and estimate their values. In the next four sections we go beyond this intuitive approach and develop tools for computing limits in a precise way. The next theorem provides our first set of tools.
The proof of Theorem 1 is discussed in Section 2.9 and Appendix D. To illustrate the underlying idea, consider two numbers such as 2.99 and 5.001. Observe that 2.99 is close to 3 and 5.0001 is close to 5, so certainly the sum 2.99 + 5.0001 is close to 3 + 5 and the product (2.99)(5.0001) is close to (3)(5). In the same way, if f(x) approaches L and g(x) approaches M as x → c, then f(x) + g(x) approaches the sum L + M, and f(x)g(x) approaches the product LM. The other laws are similar.
THEOREM 1 Basic Limit Laws
If 
and 
exist, then
-
(i) Sum Law:
exists and
-
(ii) Constant Multiple Law: For any number k,
exists and
-
(iii) Product Law:
exists and
-
(iv) Quotient Law: If
, then 
exists and
-
(v) Powers and Roots: If p, q are integers with q ≠ 0, then
exists and
Assume that 
if q is even, and that 
if p/q < 0. In particular, for n a positive integer,
In the second limit, assume that 
if n is even.
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Before proceeding to the examples, we make some useful remarks.
-
• The Sum and Product Laws are valid for any number of functions. For example,
-
• The Sum Law has a counterpart for differences:
This follows from the Sum and Constant Multiple Laws (with k = −1):
-
• Recall two basic limits from Theorem 1 in Section 2.1.4:
Applying Law (v) to f(x) = x, we obtain
for integers p, q. Assume that c ≥ 0 if q is even and that c > 0 if p/q < 0.
Use the Basic Limit Laws to evaluate:
-
(a)
-
(b)
-
(c)
Solution
-
(a) By Eq. (1),
. -
(b)
-
(c) By Law (v) for roots and (b),
Evaluate (a)
and (b)
.
Solution
-
(a) Use the Quotient, Sum, and Constant Multiple Laws:
You may have noticed that each of the limits in Examples 1 and 2 could have been evaluated by a simple substitution. For example, set t = −1 to evaluate
Substitution is valid when the function is continuous, a concept we shall study in the next section.
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-
(b) Use the Product, Powers, and Sum Laws:
The next example reminds us that the Basic Limit Laws apply only when the limits of both f(x) and g(x) exist.
Assumptions Matter
Show that the Product Law cannot be applied to 
if f(x) = x and g(x) = x−1.
Solution For all x ≠ 0 we have f(x)g(x) = x · x−1 = 1, so the limit of the product exists:
However, 
does not exist because g(x) = x−1 approaches ∞ as x → 0+ and it approaches −∞ as x → 0−. Therefore, the Product Law cannot be applied and its conclusion does not hold:
2.3.1 Summary
-
• The Basic Limit Laws: If
and 
both exist, then
-
(i)
-
(ii)
-
(iii)
-
(iv) If
, then 
-
(v) If p, q are integers with q ≠ 0,
For n a positive integer,
-
(i)
-
• If
or 
does not exist, then the Basic Limit Laws cannot be applied.
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