• The limit of a sum is the sum of the limits
• The limit of a difference is the difference of the limits
• The limit of a constant times a function is the constant times the limit of the function
• The limit of a product is the product of the limits
• The limit of a quotient is the quotient of the limits, provided the denominator is not zero

In general lim f(x) = f(lim x) as long as the function f is continuous.

Evaluate the limit of a complex expression one term at a time

Evaluate the limit of a vector, component by component.

If you end up with an expression such as 0/0, try to cancel common terms or reduce the expression to a simpler form before evaluating the limit.

Theorem lim_x->a f(x)=L exists if and only if the limit exists as x->a^- and x->a⁺

For this reason lim_x->0 |x|/x does not exist.

Theorem if f(x) < g(x) for all x on an interval (c,d) containing a, then

lim _x->a f(x) < lim _x->a g(x) Theorem if f(x) < g(x) < h(x) for all x on an (c,d) containing a, and lim_x->a f(x)=lim_x->a h(x) = L, then lim_x->a g(x) = L

The previous theorem is useful for expressions such as x sin(1/x)

Maple has a simple syntax for limits, namely limit(f,x=a); or Limit(f,x=a);

If the direction is important, one has limit(f,x=a,direction); or Limit(f,x=a,direction); where direction can be "left", "right", "real", or "complex". Examples:

• limit (x*sin(1/x),x=0);
• limit ((x^2-1)/(x-1),x=1);
• limit(abs(x)/x,x=0,left);
• limit(abs(x)/x,x=0,right);

Last modified Mon Sep 9 21:22:37 CDT 1996