clear all
; Read this file into Mathomatic to learn how the limit command does limits.
; The formula for the derivative of a function of "x" (f(x)) is:

;       f(x+h) - f(x)
; limit -------------  As "h" goes to 0.
;             h

; Here we will compute the derivative of "x^.5" using limits:

y=((x+h)^.5-x^.5)/h
limit h 0 ; Take the limit of the current equation as "h" goes to 0.
; The result should be "y = 1/(2*x^0.5)".
pause
; To show how the limit command works, we will do what it does, step by step.

1 ; Select the original equation (equation number 1).
; We want "h" to go to 0 to give us the derivative.
; Just entering 0 for "h" will give a divide by zero error.
; So we will first have to solve for "h":

h
pause
; Simplify and replace "h" with 0:

simplify symbolic
replace h with 0
pause
; Last step, solve the equation back for "y":

y
compare 1 with 2 ; Compare with the result of the limit command.

; Obviously this method only works if the equation is solvable.
pause
; INFINITY LIMITS
; ---------------
; To take the limit as some variable "h" goes to infinity,
; remember that 1/infinity is essentially 0, and the limit
; of "1/h" as "h" goes to 0 is +/-infinity, so replace "h"
; with "1/h" and take the limit as "h" now goes to 0.
; Note that this method doesn't always work and sometimes gives wrong answers.

; Let's try a simple example:
y=(5x+100-a)/(x-b)
limit x inf ; Take the limit as "x" goes to infinity.
; The limit command should say the result is 5.
pause
; To show how the limit command works, we will do what it does, step by step.

3 ; Select the original equation (equation number 3).
; To take the limit as "x" approaches infinity,
; first replace "x" with "1/x":

replace x with 1/x
pause

; Simplify and replace "x" with 0:
simplify
replace x with 0
; The result should be 5.