Description:
“Sky is the Limit” is a manipulative that will tackle some
of the lessons about limit. The purpose of this paper is to
show how infinite limits works using a 3-dimensional object.
Our 3-D object is a miniature palm tree wherein the values of
X are represented by coconuts, and the values of f(x) are
represented by the trunk. When the coconut, which
represents the values of X, goes down or decreases, the
values of f(x) also decreases. When the coconut, which
represents the values of X, goes up or increases, the values of
X also increases. Thus, values of f(x) increases or decreases as
values of X also increases or decreases without bound as it
approaches to negative or positive infinity.
Materials:






Wood
Glue sticks and glue gun
Cartolina (green)
Tie wire
Illustration board
Paint
Picture of the Manipulative:
Discussion:
Concept of Infinite Limits
In this paper, we will know how infinite limits works
using a 3-dimensional object.
Limit is the approximate values of f(x) that gets closer to
a certain number. But what happens when the function
doesn’t approach any certain f(x)? Thus, we say that the limit
does not exist. However, if F approaches infinity as X
approaches a number C, the values of f(x) becomes smaller
or larger without bound as X approaches C. This function is
said to have infinite limit. Infinite limits are those that have a
value of positive or negative infinity where the function
increases or decreases without bound.
In our 3-D model, the parts of the palm tree will
represent both values of X and Y. Wherein the coconut will
represent the values of X as X approaches to a certain value
of Y, represented by corresponding values of Y on each trunk.
As the values of X decreases, the values of Y also decreases.
As the values of X increases, the values of Y also increases.
The function
−2𝑥+𝑥
𝑥−1
will determine whether the values
of both X and Y will increases or decrease.
Example:
−2𝑥 + 𝑥
𝑥→1 𝑥 − 1
lim
x
-.9
-.99
-.999
-.9999
f(x)
1.526315789
1.502512563
1.500250125
1.500025001
x
1.1
1.01
1.001
1.0001
f(x)
-9
-99
-999
-9999
Assuming that the coconuts are the values of X and the
trunks are the values of f(x). If the decreasing values of X are
substituted to the function, the corresponding f(x) values
decreases. If the increasing values of X are substituted to the
function, the corresponding f(x) values increases.
Solution:
lim =
𝑥→1
−2𝑥 + 𝑥
𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡
𝑥−1
To get the limit from both sides,
−2𝑥 + 𝑥
lim 𝑓(𝑥 ) =
=Ꝏ
𝑥→1
𝑥−1
= -2 + 1 = -1
 As x→1, (x-1)→0 through negative values
lim 𝑓(𝑥 ) =
𝑥→1
−2𝑥 + 𝑥
= −Ꝏ
𝑥−1
= -2(1) + (1) = -1
 As x→1, (x-1)→0 through positive values
Illustration:
As the x from the left
approaches -.9, the f(x)
approaches
1.526315789.
As x from the right
approaches 1.1, the f(x)
approaches -9.
As the x from the left
approaches -.99, the f(x)
approaches
1.502512563.
As x from the right
approaches 1.01, the f(x)
approaches -99.
As the x from the left
approaches -.999, the f(x)
approaches 1.500250125.
As x from the right
approaches 1.001, the f(x)
approaches -999.
As the x from the left
approaches -.9999, the f(x)
approaches 1.500025001.
As x from the right
approaches 1.0001, the f(x)
approaches -9999.
Documentation: