- 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*^{+}.

- as
*x*approaches*c*, written*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*,

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*= −∞.

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

- lim
_{x → c}*x*=*c*.

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

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

- lim
_{D}(*u*+*v*) = ∞ if lim_{D}*u*> −∞ and lim_{D}*v*= ∞;

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*.

- ∞ − ∞; 0 · ∞; ∞ ÷ ∞; 0 ÷ 0.

- lim
_{D}(*u*/*v*) = lim_{D}(d*u*/d*v*) if lim_{D}*v*= ∞, lim_{D}*v*= −∞, or lim_{D}*u*= 0 and lim_{D}*v*= 0.

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.

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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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