Math 251 - What is a limit?
Summary
TLDRIn this Math 251 lesson, the concept of limits in calculus is introduced. The instructor breaks down the formal definition, explaining how a function's values approach a certain real number 'l' as the input 'x' approaches a specific value 'a'. Using both graphical and tabular examples, the lesson illustrates how to determine limits, even when the function is undefined at 'x=a', emphasizing the importance of the function's behavior around 'a' rather than at the point itself.
Takeaways
- 📘 The concept of a limit in calculus is introduced, focusing on how function values approach a certain number.
- 📐 A function f(x) must be defined on an interval containing a point 'a', except possibly at 'a' itself.
- 🔢 The limit 'l' is a real number that represents the y-value the function approaches as x approaches 'a'.
- 📉 As x values get closer to 'a', the corresponding f(x) values should get closer to 'l'.
- 📊 The limit is symbolized as 'lim (x→a) f(x) = l', indicating f(x) approaches l as x approaches a.
- 📈 An example is given with a quadratic graph to illustrate how to find the limit as x approaches a specific value.
- 📋 Even with a table of values, the concept of a limit can be applied to determine what the function values approach.
- 🚫 The function does not need to be defined at the point 'a'; it's acceptable to have a 'hole' at that point.
- 📌 The limit exists and is valid even if the function is undefined at the point where x is approaching.
- 👋 The video concludes with a reminder that there are more lessons to follow on this topic.
Q & A
What is the primary focus of the lesson in the transcript?
-The lesson focuses on explaining the concept of limits in calculus, specifically what it means for the limit of a function f(x) to approach a value l as x approaches a certain point a.
How is the definition of a limit presented in the lesson?
-The definition of a limit is presented as a function f(x) being defined at all points in an open interval containing a, except possibly at a itself, with the function values approaching a real number l as x values approach a.
What does it mean for a function to be defined at all values in an open interval containing a?
-It means the function is defined throughout the interval except possibly at the point a, allowing for a 'hole' at that point without affecting the limit.
Why is it okay for the function to have a hole at point a?
-Having a hole at point a is acceptable because the definition of a limit only requires the function to be defined in an interval around a, not necessarily at a itself.
What is the symbolic representation of a limit as x approaches a?
-The symbolic representation of a limit as x approaches a is written as 'lim (x→a) f(x) = l', indicating that f(x) approaches l as x approaches a.
How does the lesson use a graph to illustrate the concept of a limit?
-The lesson uses a graph to show that as x values get closer to a, the corresponding y values (f(x)) get closer to a specific real number l, which is the limit.
What is an example of finding a limit using a graph provided in the lesson?
-An example is finding the limit as x approaches 6 on a quadratic graph, where it's observed that as x values approach 6 from both sides, the y values approach 2.
How can you determine a limit when given a table of values instead of a graph?
-With a table, you look at how the y values (f(x)) change as the x values approach the point of interest from both sides to see if they consistently approach a certain value l.
What does it mean if the function is undefined at the point x equals a?
-If the function is undefined at x equals a, it means there is a hole or a discontinuity at that point, but it does not prevent the existence of a limit as x approaches a.
What is the significance of the y value in the context of limits?
-The y value is significant because it represents the value that the function f(x) approaches as x approaches a certain point a, which is the essence of the limit concept.
How does the lesson emphasize the importance of approaching a from both sides when considering limits?
-The lesson emphasizes that to find a limit, one must consider how the function values behave as x approaches a from both directions, ensuring the function's behavior is consistent.
Outlines
📘 Introduction to Limits
This paragraph introduces the concept of limits in calculus, specifically focusing on the definition of a limit from a textbook. The instructor breaks down the definition into digestible parts to make it easier to understand. It starts with defining a function 'f(x)' that is defined on an open interval containing a point 'a', except possibly at 'a' itself. The limit 'l' is introduced as a real number that the function values approach as 'x' approaches 'a'. The concept is then visualized through a graph, where 'x' values get closer to 'a' from both sides, and the corresponding 'y' values (function values) get closer to 'l'. The instructor uses a quadratic graph as an example to illustrate how to find the limit as 'x' approaches a certain value, showing that the limit can be determined by observing the 'y' values as 'x' values approach the point in question.
📐 Practical Example of Limits
The second paragraph provides a practical example of finding a limit using a table of values instead of a graph. The instructor explains how to determine the limit as 'x' approaches a certain value by looking at the 'y' values in the table. It is noted that even if the function is undefined at the exact point 'x' equals 3, the limit can still be determined and is valid, as long as the function is defined on an interval around that point. The example shows that the 'y' values approach a certain number regardless of whether the function is defined at the exact point or not. This highlights the importance of understanding that the existence of a limit does not depend on the function being defined at the point itself, but rather on the behavior of the function as it approaches that point.
Mindmap
Keywords
💡Limit
💡Function
💡Open Interval
💡Real Number
💡Approach
💡Symbolically Represent
💡Graph
💡Undefined
💡Hole
💡Table
💡Y-value
Highlights
Introduction to the concept of limits in calculus
Definition of limit from a textbook, broken down for easier understanding
Explanation of a function being defined on an open interval containing a point 'a'
Clarification that the function can have a hole at point 'a'
Definition of 'l' as a real number representing the y-value the function approaches
Description of how values of 'f(x)' approach 'l' as 'x' approaches 'a'
Graphical representation of the limit concept
Visual demonstration of how 'f(x)' approaches 'l' from both sides of 'x=a'
Symbolic representation of the limit: lim (x→a) f(x) = l
Example of finding the limit of a quadratic function as x approaches 6
Explanation of how to interpret the limit from a graph
Example of finding the limit using a table of values
Explanation of how the function can be undefined at 'x=3' but still have a valid limit
Emphasis on the importance of the function being defined on an interval, even if it excludes point 'a'
Conclusion of the lesson with a reminder of the key points
Invitation to watch more lessons on the topic
Transcripts
hi everyone welcome back to math 251
in today's lesson we are talking all
about something called a limit
and this first definition here is
straight from the textbook
it's a little bit long and complicated
so i wanted to go through it and
see what it's in piece by piece so let's
read it together and then we'll go
through it a little bit at a time
it says let f x be a function defined at
all values in an open interval
containing a
with the possible exception of a itself
and let l be a real number
if all values of the function f of x
approach the real number l
as the values of x possibly not equal to
a
approach the number a then we say that
the limit of f x as
x approaches a is l more succinct
as x gets closer to a f x gets closer
and stays close to
l symbolically we can represent this
idea as the limit as
x approaches a of f of x is equal to l
so what is all of this sand let's go
through it a little chunk at a time
it first says let f x be a function
defined at all values in an open
interval containing a
with the possible exception of a itself
so basically if
the function that we are looking at here
is f x
this piece is saying that we have to be
defined on some interval
that we are concerned with maybe some
interval like this
except it's okay if we have a hole
at the point a we just have to be
defined everywhere else
next little chunk here says that l is a
real number
well that just means exactly what it
sounds like l is a number here
the important thing to remember when we
are looking at this number l
is that l is always going to be the y
value
on the graph that we are concerned with
next
it says that if all values of f of x
approach the real number
l as the values of x approach the number
a
then we have this limit that we're
talking about so
let's um let's draw a little bit on this
graph here
it says if all values of the function f
of x
approach the real number l as the values
of x
approach the number a so we are getting
really close
in the x direction to this number a
from both sides that's what's happening
here
so that is our chunk here
as the values of x approach the number a
and and we want the y values to get
really close to our number
l so what's happening in the y direction
here
well if we follow our graph as x gets
really close
to a from this direction we can see that
on the graph yeah we are approaching
this number
l and as we approach a from the other
side of the graph
we are also getting really close to that
y value of l
so this is what it looks like on a graph
and if this is true
if all of the y values approach this
number l
as all of the x values get really close
to this value a
then we can write this limit we read
this as the limit
as x approaches a of f x is equal to
l so this is kind of a visual
representation of what a limit actually
is
um so now let's do a few examples
here we have this graph it is a
quadratic graph
and we want to know what the limit as x
approaches 6 of f of x
is well first of all we know that this
limit is going to be
a y value
y value and we want to know what the y
value of this function is
as the x value gets really close to 6.
so if i were to go on my x-axis here i'm
getting really close
to this value of 6 from both sides
so i'm approaching 6 and then what
happens on
my graph well if i approach x equals 6
from this side of the graph i see that
i'm getting really close to that y value
of 2
and if i approach x equals 6 from this
side of the graph
i also see that i get this y value of
2. so that means that the limit
as x approaches 6 of our function
f of x is equal to 2
where this 2 is the y value that
corresponds to
that ordered pair as x approaches 6.
what about a table what if we were given
this table instead
and asked to find the limit as x
approaches 3 of
f x so here we don't have
a visual representation but we can still
figure it out um again we want the y
value here
and we can see that in this table all of
our x values
on both sides a little bit less than
and a little bit greater than the value
that we care about
are all approaching the number three so
here all of our x values are getting
really really really close to three
and what's happening to our y values
well on this
side they are slightly less than the
number six
and on this side they are slightly
greater than the number six
regardless from both sides of x equals
three
our y value is approaching six so that
is our answer here
we have the limit as x approaches
3 of f of x
is equal to 6. now it's worth noting
that if we look at this graph
at x equals 3 our function is undefined
that is okay all this means is that
there is a hole
at x equals three and if we think back
to that definition that we read
at the beginning of this video it says
that we just need f to be defined on
some interval
and it's okay if that interval does not
include that point
a so here we just have that hole at x
equals three
which means that it's undefined and that
does not affect
the fact that we still have this limit
that is valid so that's pretty important
to keep in mind
that's it for this video we have a
couple more for today so go ahead and
watch those
bye
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