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An Introduction to Limits with Maple

by: Paras31 245 Maple 2024
calculus limit
March 21 2025
2  0

LIMITS

Limits in maths are defined as the values that a function approaches the output for the given input values. Limits play a vital role in calculus and mathematical analysis and are used to define integrals, derivatives, and continuity. It is used in the analysis process, and it always concerns the behavior of the function at a particular point. The limit of a sequence is further generalized in the concept of the limit of a topological net and related to the limit and direct limit in the theory category. Generally, the integrals are classified into two types namely, definite and indefinite integrals. For definite integrals, the upper limit and lower limits are defined properly. Whereas indefinite integrals are expressed without limits, and it will have an arbitrary constant while integrating the function.

Sometimes we can't work something out directly ... but we can see what it should be as we get closer and closer!

Example 1

> 

"restart; f(x):=(|x|-3)/(x-3);"

proc (x) options operator, arrow, function_assign; (abs(x)-3)/(x-3) end proc

 (1)
> 

plot(f(x), x = -10 .. 10, discont = true, color = "Green")

 
> 

f(3)

Error, (in f) numeric exception: division by zero

  

Now 0/0 is a difficulty! We don't really know the value of 0/0 (it is "indeterminate"), so we need another way of answering this.

So instead of trying to work it out for x=3 let's try approaching it closer and closer:

> 

f(3.01)

1.000000000

 (2)
> 

f(3.0000001)

1.000000000

 (3)
> 

f(2.9999999)

1.000000000

 (4)
> 

Limit(f(x), x = 3)

Limit((abs(x)-3)/(x-3), x = 3)

 (5)
> 

limit(f(x), x = 3)

1

 (6)
> 

limit(f(x), x = 3, left)

1

 (7)
> 

limit(f(x), x = 3, right)

1

 (8)

Example 2

Sometimes some functions are not continuous. That is, they appear to be approaching two different values when they are approached from two sides.

> 

"g(x):=piecewise(0<x<2,1/(2 x-x^(2)),2 <x<=3,2 -x,3<x<4,x-4, 4<=x,Pi,undefined);"

proc (x) options operator, arrow, function_assign; piecewise(0 < x and x < 2, 1/(2*x-x^2), 2 < x and x <= 3, 2-x, 3 < x and x < 4, x-4, 4 <= x, Pi, undefined) end proc

 (9)
> 

plot(g(x), x = -10 .. 10, y = -1 .. 10, discont = true, color = "Red")

 

Suppose we want to approach 2 and see the function’s limit. This naturally leads to directions from which we can approach. Left-hand side and the right-hand side limits.

The right-hand side limit is the value of the function that it takes while approaching it from the right-hand side of the desired point. Similarly, the left-hand side limit is the value of function while approaching it from the left-hand side.

> 

eval(g(x), x = 2)

undefined

 (10)
> 

limit(g(x), x = 2, left)

infinity

 (11)
> 

limit(g(x), x = 2, right)

0

 (12)
> 

limit(g(x), x = 2)

undefined

 (13)

And the ordinary limit "does not exist".

> 

g(4)

Pi

 (14)
> 

limit(g(x), x = 4, left)

0

 (15)
> 

limit(g(x), x = 4, right)

Pi

 (16)
> 

limit(g(x), x = 4)

undefined

 (17)

And the ordinary limit "does not exist".

> 

with(Student[Calculus1]); LimitTutor()

Example 3

Estimate the value of the following limit limit(h(x)*where, x = 2), h(x) = piecewise(x <> 2, x+12, x = 2, 4).

> 

"h(x):={[[x+12,x<>2],[4,x=2]];"

proc (x) options operator, arrow, function_assign; piecewise(x <> 2, x+12, x = 2, 4) end proc

 (18)
> 

plot(h(x), x = -10 .. 10, discont = true, color = "#40e0d0")

 
> 

limit(h(x), x = 2)

14

 (19)

The limit is NOT 2025!Remember from the first example that limits do not care what the function is actually doing at the point in question. Limits are only concerned with what is going on around the point. Since the only thing about the function that we actually changed was its behavior at x = 2 this will not change the limit.

Example 4

> 

" w(x):=piecewise( x<0,-x+5,x>=0,2 x);"

proc (x) options operator, arrow, function_assign; piecewise(x < 0, -x+5, 0 <= x, 2*x) end proc

 (20)
> 

plot(w(x), x = -10 .. 10, y = -10 .. 10, discont = true, color = "Blue")

 
> 

limit(w(x), x = 5)

10

 (21)
> 

limit(w(x), x = 6, left)

12

 (22)
> 

limit(w(x), x = 1, right)

2

 (23)

Example 5

> 

" k(x):=piecewise( x<5,x+4,x>=5, x^(2)-2);"

proc (x) options operator, arrow, function_assign; piecewise(x < 5, x+4, 5 <= x, x^2-2) end proc

 (24)
> 

plot(k(x), x = -10 .. 10, discont = true, color = orange)

 
> 

limit(k(x), x = 2)

6

 (25)
> 

limit(k(x), x = 5, left)

9

 (26)
> 

limit(k(x), x = 5, right)

23

 (27)
> 

limit(k(x), x = 5)

undefined

 (28)
> 

limit(k(x), x = 6)

34

 (29)

Example 6

> 

restart
> 

" l(x):=piecewise( x<=1,(x-8)/(x-3),x>=3, sqrt(x^(2)+x+2), undefined);"

proc (x) options operator, arrow, function_assign; piecewise(x <= 1, (x-8)/(x-3), 3 <= x, sqrt(x^2+x+2), undefined) end proc

 (30)
> 

plot(l(x), x = -10 .. 10, discont = true, color = "Blue")

 
> 

limit(l(x), x = 0)

8/3

 (31)
> 

limit(l(x), x = 1, left)

7/2

 (32)
> 

limit(l(x), x = 1, right)

undefined

 (33)
> 

limit(l(x), x = 2)

undefined

 (34)

Example 7

Estimate the value of the following limit. limit(H(t), t = 0)where, H(t) = piecewise(t < 0, 0, t >= 0, 1)

> 

" H(t):=piecewise( t<0,0,t>=0, 1);"

proc (t) options operator, arrow, function_assign; piecewise(t < 0, 0, 0 <= t, 1) end proc

 (35)

This function is often called either the Heaviside or step function. We could use a table of values to estimate the limit, but it’s probably just as quick in this case to use the graph so let’s do that. Below is the graph of this function.

> 

plot(H(t), t = -10 .. 10, discont = true, color = "Blue")

 
> 

limit(H(t), t = 0, left)

0

 (36)
> 

limit(H(t), t = 0, right)

1

 (37)

We can see from the graph that if we approach t = 0from the right side the function is moving in towards a yvalue of 1. Well actually it’s just staying at 1, but in the terminology that we’ve been using in this section it’s moving in towards 1.

Also, if we move in towards t = 0 from the left the function is moving in towards a yvalue of 0.

According to our definition of the limit the function needs to move in towards a single value as we move in towards t = a (from both sides). This isn’t happening in this case and so in this example we will also say that the limit doesn’t exist.

 

NULL

Download limits.mw

How remove issue of Float(undefined)?...

Asked by: salim-barzani 1575
user-error
March 21 2025
3  2

i did plot without reducing the decimal but when i reduce to 2 decimal this error is showing up How i fix this issue?
plot.mw

How do I change the background colour of a workshe...

Asked by: one pound 35
worksheet background color
March 21 2025
2  1

Looking back at this question:

https://www.mapleprimes.com/questions/37691-How-Do-I-Change-The-Background-Colour

the reply by Hmm in 2009

"In Linux there is an X11 resource file for the classic worksheet, one of the settings controls the background color.  The distributed file is located in the maple/X11_defaults directory."

suggested a solution only thiis does not work for me in Maple2024. Perhaps because this worked for the classic view.

Why is there still no way to do this after 16 years or perhaps I just cannot find the option?

Is there away to change the background colour or even invert all colours?

What type of region in my orlsheet?...

Asked by: awass 301
worksheet editing
March 21 2025
0  0

I occasionally find myself inexplicably in a region of my worksheet that does not respond to my editing, fe.g., I cannot erae or I cannot copy. I recently typed a command-there was an error of some form and I tried unsuccessfully to edit it. Then I tried to copy the code and enter it at a prompt. It refused to enter it at the prompt but copied it below the prompt in a strnge region that I do not understand. When I retyped the command at a prompt it worked fine. I do not understand th dead regions that pop up mysteriously. (I am using Maple 2024 on a Mac) The page is too long to copy but I could upload it if given directions.

geometry problems...

Asked by: yangtheary 290
geometry
March 21 2025
2  7

How do true evaluation for finding unknown?...

Asked by: salim-barzani 1575
parameters algsubs duplicate-question
March 21 2025
3  2

in this equation we have a list of a lot paramter which i have to find it but in prgress to find it some issue are appear i don't know how many term i have to replacing by algsubs there is any way for showing that something like lpring just for factoring and replacing if we did something like that in one step we can replacing all and then find our parameter there is any way for finding parameter like that?

Download pro.mw

How remove sign of integral by substitution?...

Asked by: salim-barzani 1575
substitution pde differential-equations
March 21 2025
3  4

in this integral PDE author did a substitution and the integral is simplify and removing how i can do that as mention in picture i did try but i think need a technique

Download int.mw

Understanding PDEs' Initial Condition Notations ...

Asked by: Paras31 245
March 20 2025
1  6

All three expressions define the same initial velocity condition in different notations.
ic1 := u(x, 0) = f, D[2](u)(x, 0) = g
ic2 := u(x, 0) = f, diff(u(x, 0), t) = g;
ic3 := u(x, 0) = f, u__t(x, 0) = g;  

Why does only ic1 work while ic2 and ic3 do not? Should I use another way?

wave_equation_1D.mw

Change default entry style for execution cells?...

Asked by: studentX369 30
March 20 2025
1  7

Working on updating a very large and out of date guidebook and I need the content of execution cells to be 'Text' for illustrative purposes. It's tedious to constantly have to hit F5 for every, single, new, cell as I'm working through this. Is there a way I can just have the default style be 'Text' instead of 'Math'?  I can't seem to find that as an option anywhere.

Inserting a new text prompt shortcut key?...

Asked by: studentX369 30
March 20 2025
1  3

Ctrl-j/k work fine for inserting a new execution prompt, why doesn't there seem to be a short cut for a new text prompt?  Ctrl+shift+j/k just enters a new line in the current execution prompt.  I'm on Linux using Maplesoft 2024.

How to solve term7?...

Asked by: Alfred_F 335
March 20 2025
1  7

I'm doing something wrong again in the attached file. Please advise.

test.mw

Plotting problem...

Asked by: MichaelVio 40
March 19 2025
0  11

Hi everyone,

I'm having some trouble with plotting in Maple and was hoping to get some help here. I'm trying to create a plot for a specific function, but I'm not sure if I'm using the right commands or parameters. Here’s what I’m trying to do:

To plot the line plot(E3,nu=10^13..4.5*10^18);

and 

plot(E1(nu),nu=10^13..4.5*10^18); 

you can look around my calculus please advise.

Could someone please explain what might be going wrong with my approach? Any suggestions or examples of correct usage would be greatly appreciated.

Thanks in advance for your help!PLanckPh.mw

Generate Similar with a table layout...

Asked by: jalal 435
March 19 2025
1  1

Hi,

I am trying to create a series of exercises ( Generate Similar) with a table layout (2x3), but my approach is not optimal. Any suggestions? Thanks

GenerateSimilar_Test.mw

How to lower the plot title? ...

Asked by: C_R 3432
March 19 2025
2  7

Is it possible to reduce the space between the plot title and the 3dplot in the attached file?

Plot_title_too_high.mw

fyi, Maple 2025 new interface video

by: nm 11353 Maple
interface
March 13 2025
1  2

fyi, there is new video showing Maple's 2025 new interface

It seems oriented to document mode which I do not use. May be they also improved worksheet mode.

I am still getting my Maple desktop getting shuffled few times each day where I have to close Maple and reopen it to clear it. I hope they fixed this in the new interface.,

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