- as
*x*increases without limit:*x*→ ∞; - as
*x*decreases without limit:*x*→ −∞ - as
*x*increases towards*c*:*x*→*c*^{−}; - as
*x*decreases towards*c*:*x*→*c*^{+}.

- as
*x*approaches*c*:*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*.

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:

- 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*= −∞.

- lim
_{x → ∞}*x*= ∞; - lim
_{x → −∞}*x*= −∞; - lim
_{x → c−}*x*=*c*; - lim
_{x → c+}*x*=*c*.

- lim
_{x → c}*x*=*c*.

- lim
_{D}*C*=*C*.

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 automatically also defined):

- lim
_{D}(*u*+*v*) = lim_{D}*u*+ lim_{D}*v*; - lim
_{D}(*u*−*v*) = lim_{D}*u*− lim_{D}*v*; - lim
_{D}(*u**v*) = lim_{D}*u*· lim_{D}*v*; - lim
_{D}(*u*/*v*) = lim_{D}*u*÷ lim_{D}*v*.

- lim
_{D}(*u*^{C}) = (lim_{D}*u*)^{C}; - lim
_{D}(^{C}√*u*) =^{C}√(lim_{D}*u*).

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 *C* is a (finite) constant:

*a*+ ∞ = ∞ if*a*> −∞;*a*− ∞ = −∞ if*a*< ∞;*a*· ∞ = ∞ if*a*> 0;*a*· ∞ = −∞ if*a*< 0;*a*÷ ∞ = 0 if −∞ <*a*< ∞;- ∞
^{C}= ∞ if*C*> 0; ∞^{C}= 0 if*C*< 0; ^{C}√∞ = ∞ if*C*> 0.

- 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*.

- ∞ − ∞; 0 · ∞; ∞ ÷ ∞; 0 ÷ 0.

- lim
_{D}(*u*/*v*) = (lim_{D}d*u*) ÷ (lim_{D}d*v*) if the right-hand side is defined and lim_{D}*u*and lim_{D}*v*are either both 0 or both ±∞.

This rule shows the connection between limits and differential calculus. In the official textbook, this connection is exploited in the other direction, to define derivatives (and so essentially differentials) using limits.

Go back to the course homepage.

This web page was written in 2011 by Toby Bartels, last edited on 2011 June 9. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-1400/2011s/limits/`

.