> limit(expression or function, x=c);Notice that the first argument in the parentheses is the expression or function that you are taking the limit of; the second is the value that is approaching. You can type the expression directly into the limit command or another approach is to first define the expression or function with a label in Maple and use the variable name within the

> limit(x^2+8*sin(-3*Pi*x), x=1/2); > f:=x->(cos(x)-1)/x; > limit(f(x),x=0);If the limit exists, Maple can usually find it. In cases where the limit doesn't exist, such as or where , Maple gives the answer undefined. Look at the plots of the functions to see why the limits are undefined.

> limit(1/x,x=0); > plot(1/x,x=-5..5,y=-10..10); > p := piecewise(x<0,-x^2+4,x>=0,x-2); > plot(p,x=-5..5); > limit(p,x=0);Another reason that a limit may not exist at a particular point is because of oscillations. For instance, the limit as approaches 0 of is undefined. To get a better idea of why, look at the plot. When evaluating the limit using Maple, the result is the range -1..1. When the limit doesn't exist, but the expression or function is bounded, this is the type of answer that Maple will give.

> plot(cos(1/x),x=-2..2); > limit(cos(1/x),x=0);Maple also can find one-sided limits. Suppose that you want to find the limit as approaches 0 from the right for the function .

> f:=x->x*sin(1/x); > limit(f(x),x=0,right);Similarly, you can evaluate one-sided limits from the left by replacing the second argument with the word

When using the definition to compute a derivative in Maple, it is easiest to first define the function . You can then find the difference quotient of and take the limit as approaches 0 in either one or two steps. Both examples are shown below.

> f := x -> x^2+3*x+5; > DQ:=(f(x+h)-f(x))/h; > derivative:=limit(DQ,h=0); > derivative:=limit((f(x+h)-f(x))/h,h=0);To evaluate the derivative at a given x value, you can use one of two methods. Both are given below for .

> subs(x=1,derivative); > limit ((f(1+h)-f(1))/h,h=0);

- Use Maple to evaluate each of the limits given below. If the limit exists, state the limit. If the limit does not exist, explain why. A plot may be necessary to support your answer.
- Find the derivative of each function below using the definition of the derivative. Also, evaluate the derivative at the given value.
- ,
- ,

2001-09-24