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 xc, 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.


Before proceeding to the examples, we make some useful remarks.

Use the Basic Limit Laws to evaluate:

  • (a)
  • (b)
  • (c)


  • (a) By Eq. (1), .
  • (b)
  • (c) By Law (v) for roots and (b),

Evaluate (a) and (b).


  • (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 evaluateSubstitution is valid when the function is continuous, a concept we shall study in the next section.


  • (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