The document discusses the concept of limits in calculus, emphasizing their importance for understanding function behavior, particularly when approaching critical points. It provides an example using the function f(x) = (x-1)/(x-1), highlighting an undefined value at x = 1 and the implications for continuity. The document explains that limits indicate the value a function approaches as x nears a specific point, considering both left-hand and right-hand limits.

1. MATH HUB
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2. LIMITS
An important concept that make calculus possible!

3. MOTIVATION
•Imagine a line containing infinitely
many points. Since it is a line, we do
not question ourselves whether it is
totally continuous all through out.
What if there are instances where
some points of this line does not
exist? Making the line reach its
limit/end.
Lines!

4. LETS EXAMINE
•Consider the function
•𝑓 𝑥 =
𝑥−1
𝑥−1
Some functions

5. LETS EXAMINE
The table of values
𝑓 𝑥 =
𝑥 − 1
𝑥 − 1
1 1 1 1 1 1 0/0 1 1
𝑥 -5 -4 -3 -2 -1 0 1 2 3

6. LETS EXAMINE
The Graph

7. ANALYSIS • We get an undefined value 0/0 when x is equal to 1 and get
a value of 𝑓 𝑥 = 1 for other values of x. this means that at x
= 1 the function is not defined or has no meaning. Also at
that point the function is not continuous.
Lines!

8. INTERPRETATION
• Limit of a function could mean the last value outputted by
𝑓(𝑥) which is defined before it reaches a critical value when
𝑥 → 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑒𝑠 𝑎 that will yield the function being
discontinuous.
• lim
𝑥→𝑎
𝑓 𝑥 = 𝐿
• We can see that as x approaches the critical value 𝑎 , the
value gets larger and larger and may either be towards the
positive infinity or the negative infinity. Also, apparently x
approaches 𝑎 from positive infinity or negative infinity and
we can write;
• lim
𝑥→𝑎+
𝑓 𝑥 = 𝐿
• lim
𝑥→𝑎−
𝑓 𝑥 = 𝐿
Limits!