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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

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