Limits

Although limits are not the foundation in our approach to calculus, they are still an important concept. Roughly speaking, a limit describes what value one quantity is approaching as another quantity approaches some value in some direction.

Directions

A direction in some variable describes not only whether the variable is increasing or decreasing (that is its literal direction on a number line) but also if there is a limiting value that it approaches (but does not reach). The basic directions that we study in this course take the following four forms, where x may be any variable and c may be any constant:

as x increases without bound, written x → ∞;
as x decreases without bound, written x → −∞;
as x increases to c, written x → c⁻;
as x decreases to c, written x → c⁺.

Any two or more of these directions may be combined, but the only type of combined direction that we will need is this:

as x approaches c, written x → c.

This is the combination of x → c⁻ and x → c⁺.

If D is any direction and u is any variable quantity, then we indicate the value to which u approaches as change occurs in the indicated direction as

lim_D u,

called the limit of u in the direction D. We will examine the case when u approaches a real value L, as well as the case when u increases without bound or decreases without bound. In the first case, we say that the limit converges; in the second case, we say that the limit diverges to (positive or negative) infinity. Other types of behaviour are also possible, which are also kinds of divergence; in this course, we will consider those limits to be undefined.

A limit as x approaches c exists (as one of the three kinds of results that we have defined) if and only if the limit as x increases to c and the limit as x decreases to c both exist and are the same result.

So in total, there are fifteen kinds of limits that we will consider, for the five kinds of directions (four basic and one combined) and the three kinds of answers (one convergent and two divergent):

lim_x → ∞ u = L; lim_x → ∞ u = ∞; lim_x → ∞ u = −∞;
lim_{x → −∞} u = L; lim_{x → −∞} u = ∞; lim_{x → −∞} u = −∞;
lim_x → c⁻ u = L; lim_x → c⁻ u = ∞; lim_x → c⁻ u = −∞;
lim_x → c⁺ u = L; lim_x → c⁺ u = ∞; lim_x → c⁺ u = −∞;
lim_x → c u = L; lim_x → c u = ∞; lim_x → c u = −∞.

To see how to read these aloud, I'll consider the last one as an example: the limit of u, as x approaches c, is negative infinity; or the limit, as x approaches c, of u is negative infinity.

Evaluating limits

The first fact to know about limits is that the limit of the variable itself is already given by the direction:

lim_x → ∞ x = ∞;
lim_{x → −∞} x = −∞;
lim_x → c⁻ x = c;
lim_x → c⁺ x = c.

Since the limit of x is c whenever x increases or decreases to c, the limit is still c whenever x approaches c:

lim_x → c x = c.

A similarly important principle is that the limit of a constant, in any direction, is that constant:

lim_D k = k.

Of course, we rarely bother with limits as simple as these! However, we have the powerful principle that if an expression is built using only the basic operations of addition, subtraction, multiplication, and division, then the limit of the expression may be computed using these operations. Explicitly, each of these equations is true whenever the right-hand side is defined (so that in particular the left-hand side is necessarily also defined):

lim_D (u + v) = lim_D u + lim_D v;
lim_D (u − v) = lim_D u − lim_D v;
lim_D (uv) = lim_D u · lim_D v;
lim_D (u/v) = lim_D u ÷ lim_D v.

Furthermore, the same principle applies to exponentiation and taking roots, as long as the exponent or index is a constant:

lim_D (u^k) = (lim_D u)^k;
lim_D (^k√u) = ^k√(lim_D u).

If you think about these rules in detail every time that you apply them, then you're working too hard. The point is simply that you can plug in the target c of the direction for the variable x and do ordinary arithmetic, as long as the result of this arithmetic is defined. In this way, we can evaluate most limits.

We can do even more limits if we extend arithmetic to the values ±∞ as follows, where a is (in general) any real number or ±∞ and k is a constant real number:

a + ∞ = ∞ if a > −∞;
a − ∞ = −∞ if a < ∞;
a · ∞ = ∞ if a > 0; a · ∞ = −∞ if a < 0;
a ÷ ∞ = 0 if −∞ < a < ∞;
∞^k = ∞ if k > 0; ∞^k = 0 if k < 0;
^k√∞ = ∞ if k > 0.

Technically, what these statements mean is:

lim_D (u + v) = ∞ if lim_D u > −∞ and lim_D v = ∞;

and so on. But usually it's easier just to imagine calculating with infinity.

Finally, we can even divide by zero sometimes, if we are computing limits! This is trickier, so I'll state the rules more carefully:

lim_D (u/v) = ∞ if lim_D u > 0, lim_D v = 0, and v > 0 in the direction D;
lim_D (u/v) = −∞ if lim_D u > 0, lim_D v = 0, and v < 0 in the direction D;
lim_D (u/v) = −∞ if lim_D u < 0, lim_D v = 0, and v > 0 in the direction D;
lim_D (u/v) = ∞ if lim_D u < 0, lim_D v = 0, and v < 0 in the direction D;
lim_D (u/v) is undefined if lim_D u ≠ 0, lim_D v = 0, and u/v takes both positive and negative values in the direction D.

In other words, if v has a consistent sign while it approaches zero, then the limit of u/v is plus or minus infinity, depending on how the sign of v compares to the sign of u. However, this tells us nothing if lim_D u = 0 too; for that, we need the next idea.

L'Hôpital's Rule

We're left with the following indeterminate forms:

∞ − ∞; 0 · ∞; ∞ ÷ ∞; 0 ÷ 0.

Limits of these forms can usually be evaluated by using some algebra to rewrite the expression in a more amenable way. However, another approach to the last two forms is given by L'Hôpital's Rule, assuming that the right-hand side is defined:

lim_D (u/v) = lim_D (du/dv) if lim_D v = ∞, lim_D v = −∞, or lim_D u = 0 and lim_D v = 0.

So basically, we use L'Hôpital's Rule for 0/0 and for ∞/∞. Sometimes the other two indeterminate forms can also be rewritten as 0/0 or ∞/∞ using some algebra.

This rule shows the connection between limits and derivatives. In the official textbook, this connection is exploited in the other direction, to define derivatives using limits.

Exponential and logarithmic limits

I will save for later how we handle expressions involving logarithms or variable exponents. But even for these, most of the time, you can just substitute for the variable using the direction. There are some additional indeterminate forms, but they can be turned into expressions subject to L'Hôpital's Rule using some standard algebraic tricks.

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This web page was written between 2011 and 2015 by Toby Bartels, last edited on 2015 August 3. Toby reserves no legal rights to it.

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