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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);
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);
> f(x);
> limit(f(x),x=infinity);
> limit(f(x),x= -infinity);
Definition 342
We say that the number L is the limit of f(x) as x approaches a provided that, given any number , there exists a number such that
for all x such thatThis definition may seem complicated, but its graphical interpretation is not so bad. It says that if you plot f(x) - L with the y range set to you can always choose a value of small enough so that when you shrink the x plot range to and plot the function, its graph will not intersect the top or the bottom edges of your plot. For example, suppose f(x)=x2, a=2 and . Then any value of smaller than about 0.049 will work. To see what is going on, look at the plots generated by the following commands.
> 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 0.1. This is too large, since the graph intersects the lines y=-0.2 and y=0.2. The value of 0.048 for in the second plot command, however, is small enough, since the graph of f(x) goes off the sides of the plot. Make sure that you understand this example. If you don't understand, ask for help.
> plot(floor(x),x=0..4);
> limit(floor(x),x=1,right);
> limit(floor(x),x=1,left);
> f1 := x -> sqrt(2-x);
> limit(f1(x),x=2,left);
> f2 := x -> x;
> limit(f2(x),x=2,right);
Christine Marie Bonini