The purpose of this lab is to use Maple to become more familiar with limits of functions, including one-sided limits.
> 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. See the examples below.
> limit(1/x,x=0); > limit(sin(1/x),x=0);You can also use Maple to compute limits as goes to as shown below.
> f(x); > limit(f(x),x=infinity); > limit(f(x),x= -infinity);
> f := x -> x^2; > limit(f(x),x=2); > plot({-0.2,0.2,f(x)-4},x=2-0.1..2+0.1,y=-0.2..0.2); > plot({-0.2,0.2,f(x)-4},x=2-0.048..2+0.048,y=-0.2..0.2);In the first of the two plot commands, the value of is . This is too large, since the graph intersects the lines and . The value of for in the second plot command, however, is small enough, since the graph of goes off the sides of the plot. Make sure that you understand this example. If you don't understand, ask for help.
If you want to define your own
piecewise-defined function, then the Maple piecewise command
is the best way to do it. Suppose you wanted to define the following
function.
> g := x -> piecewise(x < 0, -x, x^2+1);If you want to see your function in a more familiar form, just run a command like the one below.
> g(x);The way the piecewise command works is that you give it a sequence of pairs of conditions and formulas that define your function. When you want to evaluate your function at a particular value of , Maple checks the conditions from left to right until it finds the one that your value of satisifies. It then plugs the value of into the next formula. However, notice that the command above only has one condition and two formulas. This is because any value of is either less than zero or it is greater than or equal to zero, so if a particular value of fails the first condition, i.e. is not less than zero, it must be greater than or equal to zero and the second formula is the one to use. For more information, see the help page for piecewise.
The limit command works fine for functions that are defined via the piecewise command, as shown in the example below.
> limit(g(x), x=0); > limit(g(x),x=0, left); > limit(g(x),x=0, right); > plot(g(x), x=-0.1..0.1);