limit (and Limit)

The limit command is used to compute limits (what else?). The syntax of limit is Maple's usual

limit(what,how);

syntax. "What" in this case refers to "take the limit of what?", and "how" to "as what variable approaches what value?".  For example, to compute

we would enter:

>

restart:

>	limit((3*x-6)/(x^2-4),x=2);

Notice, the first argument of limit is the expression whose limit is being taken, and the second tells which variable approaches what number.  If the limit does not exist, as in

>	limit(1/x,x=0);

Maple tells you.

Occasionally, Maple will be unable to determine a limit (or whether it exists). In such a case, Maple will return nothing -- when this happens, you can give Maple an assist sometimes by setting a variable called "Order" equal to a whole number somewhat bigger than 6 (its default variable) -- this is because Maple uses series to compute limits (you will see or have seen how to do this at the end of  Math 104), and Maple usually uses only the first six terms of the series. When you change Order, you enable Maple to do more accurate (although more time-consuming) calculations. It is somewhat like computing to more decimal places.

The limits Maple takes are "two-sided, real limits". This means Maple assumes that when you type x=15 as the second argument of limit, you mean x should approach 15 from either above or below through real values only (as opposed to complex ones). Maple can compute one-sided and complex limits. For example

>	limit(1/x,x=0,right);

You could also do limits from the left.
It is possible to compute limits as x tends to infinity (or -infinity):

>	limit(arctan(x),x=infinity);

Finally, limits of functions of more than one variable are allowed (for Math 114, 115 and 240):

>	limit(x/(x^2+y^2),{x=0,y=0});

Remarks:  It is always a good idea to use plot in conjunction with limit -- you can often get an intuition for the limit from the plot.

It is instructive to use limit for computing derivatives and integrals from the definition.

Occasionally, to make your worksheets easier to read, you may wish to have Maple display a limit in standard mathematical notation without evaluating it. For this there is a capitalized, "inert" form of the limit command:

>	Limit(exp(x)/(1-x),x=0);

Sometimes, you can use the two forms together to produce meaningful sentences:

>	Limit(exp(x)/(1-x),x=0)=limit(exp(x)/(1-x),x=0);

Few things can go wrong using the limit command, other than syntax errors -- except possibly that sometimes the variable in the limit command (the x above) has already been given a value that you forgot about:

>

x:=3;

>	limit(x^2/sin(2*x^2),x=0);

Error, (in limit) invalid limiting point

Here, limit  is objecting to your trying to let 3 approach 0.