Algebraic Limit Laws

The algebraic limit laws for finite limits say:

The Law The Precise Statement of the Law

The limit of a sum is the sum of the limits.
lim (f + g)(x) = lim f(x) + lim g(x)

x --> a x --> a x --> a

if both limits on the right-hand side exist and are finite.

The limit of a product is the product of the limits.
lim (f * g)(x) = lim f(x) * lim g(x)

x --> a x --> a x --> a

if both limits on the right-hand side exist and are finite.

The limit of a quotient is the quotient of the limits.
lim (f / g)(x) = lim f(x) / lim g(x)

x --> a x --> a x --> a

if both limits on the right-hand side exist and are finite and the limit in the denominator is not zero.

Why is THAT true?

Example: Here is a typical computation which uses the algebraic limit laws explicitly. In the final step, we use the continuity of the functions x², sin(x), and e^x. Technically, we are using the composition law to conclude that sin(x) is continuous, but at some point it becomes ridiculous to break things down to first principles.

lim	[x² + sin(x)] _/ e^x	=	[lim	x² +	lim	sin(x)] _/	lim	e^x =
x --> 1			x --> 1		x --> 1		x --> 1

[1² + sin(*1)] _/ e¹ = [1 + sin()]/e = [1 + 0]/e = 1/e.

Equipment Check: Compute the following limit and check all the boxes corresponding to facts or laws used in your computation.

The above limit laws allow us to use algebra to break limits apart. But sometimes we use algebra inside the limit. For example, in evaluating the "speed limit"

	(10 + h)² - 100
lim
h --> 0	h

we used algebra to expand the numerator and then cancel an h from the numerator an denominator. This had the effect of replacing the original function, which was not defined at h=0, by an algebraically equivalent function which included h in its domain of definition. The next law says this is permissible:

Extending the domain does not change the limit. Suppose f(x) is defined at every point (except possibly at x=a) of an open interval containing a. Suppose further that g(x) is a function whose domain contains the same open interval. Then
lim f(x) = lim g(x).

x --> a x --> a

Why is THAT true?

Example: Let's compute

	u - 1
lim		?
u --> 1	sqrt(u) - 1

The algebraic trick used here is called rationalizing the denominator.
Note that both the numerator and the denominator approach zero as u approaches 1, so we cannot write this as the quotient of two limits. To discover the limit, we need an algebraic trick. We want to rid ourselves of the square root in the denominator, and to do this we multiply the numerator and denominator by sqrt(u) + 1, noting that (sqrt(u) - 1)(sqrt(u) + 1) = u - 1.

	u - 1			(u - 1)	(sqrt(u) + 1)
lim		=	lim			=
u --> 1	sqrt(u) - 1		u --> 1	(sqrt(u) - 1)	(sqrt(u) + 1)

	~~(u - 1)~~(sqrt(u) + 1)
lim		=	lim	sqrt(u) + 1	= 2
u --> 1	~~(u - 1)~~		u --> 1

Finally, we often use the continuity of function in the Field Guide. We do not intend to give proofs of continuity for these functions, but we do wish to point out that the algebraic limit laws can be used to prove the continuity of polynomial functions. To see how, follow the Trail of Details!

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The continuity of cos(x).
"The limit of a product is the product of the limits".
A composition law.