The document contains learning objectives and examples for teaching students about limits of functions. It includes: 1) Examples of finding the limit of functions as x approaches specific values using tables of values. This shows that as x gets closer to the given value, f(x) gets closer to the limit value. 2) A definition of limit that states the limit of a function f(x) as x approaches a is L if, given any positive epsilon, there exists a delta such that if the difference between x and a is less than delta, the difference between f(x) and L will be less than epsilon. 3) Three theorems about limits: the limit of a sum or difference is the

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Teacher

2. At the end of the day, the students should be
able to:
1) Illustrate the limit of a function using table of
values and graph of the function;
STEM_BC11LC-IIIa-1
2) Distinguish between lim
𝑥→𝑐
𝑓(𝑥) and 𝑓(𝑐);
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3) Illustrate the limit laws;
STEM_BC11LC-IIIa-3
4) Apply the limit laws in evaluating the limit of
algebraic functions (polynomial, rational, and
radical).
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Find the value of 𝑓(𝑥) given the specified value of 𝑥.
1) 𝑓 𝑥 = 3𝑥 − 5 𝑤ℎ𝑒𝑛 𝑥 = −3
2) 𝑓 𝑥 =
𝑥2+5𝑥−3
𝑥+2
𝑤ℎ𝑒𝑛 𝑥 = 2
3) 𝑓 𝑥 = 2𝑥2 + 1 𝑤ℎ𝑒𝑛 𝑥 = 2.99
4) 𝑓 𝑥 = 2𝑥 − 7 𝑤ℎ𝑒𝑛 𝑥 = 3.8
5) 𝑓 𝑥 =
𝑥 − 3, 𝑥 < 1
𝑥2 + 5, 𝑥 ≥ 1
, 𝑓𝑖𝑛𝑑 𝑓(3)
-14
2.75
18.8802
0.6
0
14

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Complete the table of values representing the limit of
a function. Compare and analyze the table of values.
lim
𝑥→2
(𝑥 + 1) = 3
𝑥 𝑓(𝑥)
1 2
1.5 2.5
1.9 2.9
1.99 2.99
1.999 2.999
1.9999 2.9999
1.99999 2.99999
𝑥 𝑓(𝑥)
3 4
2.5 3.5
2.1 3.1
2.01 3.01
2.001 3.001
2.0001 3.0001
2.00001 3.00001

5. Guide Questions:
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1) How did you find the values for 𝑓(𝑥)?
2) What did you notice about the given
values of 𝑥 in the two tables?
3) What did you observe about the values
on 𝑓(𝑥)?
4) By synthesizing your observations on the
values of 𝑥 and 𝑓(𝑥) on the two tables, how
are you going to define a limit?

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Definition:
Let 𝒇 be a function that is defined at every
number in some open interval containing 𝒂,
except possibly at the number 𝒂 itself. The
limit of 𝒇(𝒙) as 𝒙 approaches 𝒂 is 𝑳, written
as 𝒍𝒊𝒎
𝒙→𝒂
𝒇 𝒙 = 𝑳, if the following statement is
true: Given any ∈ > 𝟎, however small, there
exist 𝛿 > 0 such that if
𝟎 < 𝒙 − 𝒂 < 𝜹, then 𝒇(𝒙) − 𝑳 < ∈.

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1) lim
𝑥→2
𝑥 + 1 = 3
This reads as “The limit of (x+1) as x approaches 2 is equal to 3”.
𝑥 𝑓(𝑥)
1 2
1.5 2.5
1.9 2.9
1.99 2.99
1.999 2.999
1.9999 2.9999
1.99999 2.99999
𝑥 𝑓(𝑥)
3 4
2.5 3.5
2.1 3.1
2.01 3.01
2.001 3.001
2.0001 3.0001
2.00001 3.00001

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lim
𝑥→−1
𝑥2 + 1
𝑥 𝑓(𝑥)
-1.2 2.44
-1.1 2.21
-1.01 2.0201
-1.001 2.002001
𝑥 𝑓(𝑥)
0 1
-0.9 1.81
-0.99 1.9801
-0.999 1.998001
2) Investigate by constructing table of
values then graph.

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𝑥 𝑓(𝑥)
3.7 4.7
3.85 4.85
3.995 4.995
3.9999 4.9999
𝑥 𝑓(𝑥)
4.3 3.09
4.1 3.01
4.001 3.001
4.0001 3.0001
3) Investigate through a table of values given,
lim
𝑥→4
𝑓 𝑥
𝑻𝒉𝒆𝒓𝒆𝒇𝒐𝒓𝒆, 𝐥𝐢𝐦
𝒙→𝟒
𝒇 𝒙 𝑫𝑵𝑬

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Theorem 1: Let c, k and M be real numbers, and let f(x)
and g(x) be functions defined on some open interval
containing c, except possibly at c.

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Theorem 2: Let f be a polynomial of the form
𝑓 𝑥 = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + 𝑎𝑛−2𝑥𝑛−2+. . +𝑎1 + 𝑎0
If c is a real number, then
Theorem 3: Let h be a rational function of the form
where f and g are polynomial functions. If c is a real number
and g(c) ≠ 0, then

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