Math 1350 - Limits and continuity

 

MAPLE has a powerful capability to find a limit. The MAPLE function that is used to find a limit is limit.

 

Example1:

 

find the limit of f(x) = (x-2)/(x^2-1)

 

First define the function f(x)

> f:=x->(x-2)/(x^2 - 1);

Find the limit of f(x) as x approaches 0, 1, -1, infinity

 

> limit(f(x), x=0);

> limit(f(x), x=1);

> limit(f(x), x=-1);

> limit(f(x), x=infinity);

When x approaches 1, -1 the limit does not exist therefore MAPLE gives undefined.  

 

To convince ourselves about the function´s behavior we can plot it.

> plot(f(x), x=-10..10, y=-10..10);

 

Example2:

 

a different behavior has the limit of f(x) = (x-2)/(x-1)^2.

> f:=x->(x-2)/(x - 1)^2;

> limit(f(x), x=0);

> limit(f(x), x=1);

> limit(f(x), x=-1);

> limit(f(x), x=infinity);

> plot(f(x), x=-10..10, y=-10..10);

 

Example3:

 

find the limit of f(x)=|x| as x -> 0. To write in MAPLE the absolute value of x, use the command abs.

 

> f:=x->abs(x);

> limit(f(x), x=0);

 

Example3:

 

define f(x) = sin(1/x). Find the limit as x -> 0, and as x -> infinity. Plot the function.

 

> f:=x->sin(1/x);

> limit(f(x), x=0);

> limit(f(x), x=infinity);

> plot(f(x), x=-10..10, y=-1.5..1.5);

f(x) is not continuous at as x -> 0 because its limit is ?1..1. It is continuous everywhere else since limit f(x) = 0 as x -> .

 

Exercises:

 

Find the limit of the fallowing functions a) as x -> 0; b) as x->. Plot them.

  1. f(x) = (e^x 1)/x
  2. f(x) = (e^x 1)/e^x
  3. f(x) = 5-2*x
  4. f(x) = ln(x)
  5. f(x) = cos(x)
  6. f(x) = (1/x)*cos(x)
  7. f(x) = (1/x)*sin(1/x)