> limit(x^2+2*x,x=2); > limit(sin(x)/x,x=0); > f := x -> (x+3)/(x^2+7*x+12); > limit(f(x),x=-3); > limit(f(x),x=-4);
If the limit exists, Maple can usually
find it. In cases where the limit doesn't exist, Maple gives the
answer undefined
or sometimes infinity
for an unbounded
limit or gives a range like
-1..1
if the limit doesn't exist, but the expression or
function is bounded.
You can also calculate one sided limits in Maple as shown in the example below.
> g:=x->(x-2)/(x^2-4) > g(2); > limit(g(x),x=2,left); > limit(g(x),x=2,right); > limit(g(x),x=2);
> f:=x-> x^3; > quot := (f(x+h)-f(x))/h; > der:=limit(quot,h=0); > subs(x=-2,der);or if you don't need to see that the derivative of is and you just want to evaluate the derivative at , then this could be done all in one Maple command as in the following example:
> limit((f(-2+h)-f(-2))/h,h=0);
When you use the D operator to compute the derivative of a function, the result is also a function, as shown below.
> D(f);If you provide a label, then you get a function you can use later in the session,
> df := D(f);However, this is usually not necessary. See the examples below.
If you want to evaluate the derivative at a specific value of or just get the expression for the derivative, you can use the following forms of the D operator.
> D(f)(2); > D(f)(x);This last form is the one to use for plotting, as shown below.
> plot(D(f)(x),x=-10..10);
The D operator cannot be used on expressions, for example trying to use it to differentiate the expresssion we defined above results in an error.
> D(p);If you recall that Maple uses f(x) to refer to the expresssion that is used to define , then the following error shouldn't surprise you.
> D(f(x));
To differentiate expressions, you need to use the diff command. Here is an example.
> diff(p,x);The diff command can also be applied to functions as shown below.
> diff(f(x),x);Note, however, that the result of the diff command is an expression, not a function. This means that computing the value of the derivative at a specific value of requires you to use the subs command as follows:
> der := diff(p,x); > subs(x=Pi/2,der);
Suppose you want to find where the tangent line to the graph of is horizontal. This can be done first by plotting the derivative to see how many times the graph crosses the -axis, then using Maple's solve or fsolve commands as shown below.
> f:=x->x^4-16*x^2+x/4; > plot(D(f)(x),x=-5..5); > solve(D(f)(x)=0,x) > fsolve(D(f)(x)=0,x)