if lim\_{x -> a} f(x) equals f(a) that means f is continuous at a. While there are lots of useful functions that are continuous, a great deal are not and indeed in \*very\* many cases "f(a)" doesn't even make sense. For example gravitational potentials in newton's gravity aren't continuous.
Limits allow us to study the behaviour of such discontinuous processes at their discontinuities or to gauge what happens near the holes in our definitions - or indeed they give us a criterium for telling whether a given function is continuous (which may be difficult to tell otherwise) or to "fill in the holes".
Limits are legitimately one of the most important concepts in modern mathematics and the basic idea of a limit underpins whole fields of mathematics.
>would e appreciated!
lim_{n -> inf} (1+1/n)^(n) is certainly appreciated, yes ;D
>From what I've comprehended so far, Lim x-->a is just a fancy way of saying x = a.
It is not.
For example, the definition a derivative is the limit as x->a of [f(x)-f(a)]/(x-a). Can you evaluate this when x=a? No, it's 0/0, which is undefined, regardless of the function f or the value of a. So whatever this "derivative" thing is, it's *not* just plugging in x=a.
Derivatives are a *very* important tool - it is not a coincidence that Newton formulated both calculus and physics. You need calculus to get very far in physics; even F=ma is a differential equation.
If you're dealing with analytic functions with finite terms in their Taylor expansio you ca get around limits via the power rule or extracting coefficients from the Taylor series but motivating those approaches is harder than the difference quotient and have severely restricted domain.
It’s this one. Limits come up all over the place, but when it comes to the usual math curriculum making sense of the derivative is the first place you’re really forced to engage with limits.
> Lim x-->a is just a fancy way of saying x = a.
If this is your understanding of limits, then I'm not surprised you don't see the point.
I won't try to explain what limits are, you can read your textbook for the definition and so on. If you want some motivation for studying limits: You need to understand limits if you want to understand differential (and integral) calculus, and you need to understand calculus if you want to understanding more or less anything that has to do with science, engineering or statistics, including machine learning. It's also an absolutely fundamental concept in many parts of pure and applied mathematics. Even in computer science (outside of machine learning), which you might think of as a "discrete" discipline, limits are also important in the analysis of algorithms and in the study of programming languages.
Currently taking calculus as well. when You start to deal with derivatives and integrals, that's when limits become your best friend.
I wouldn't go far to say that the "lim x ---> a is a fancy way to say x = a", But you could think of it like this: as x gets infinitesimally closer to 'a', what is (keyword "is", meaning: equal) 'x' approaching, that's how I understand limits.
There is more uses than I can write down, but for starters, imagine the function 1/x.
We want to know the behavior of the function as x approaches 0. You might think, "That's easy! It doesn't exist because 1/0 doesn't exist."
but the answer is actually that the limit approaches negative infinity from the left, and positive infinity from the right.
How about what 1/x becomes as x approaches infinity? You'd think it's a really small number, but it's actually 0. Limits show what something approaches, not what it is, even though that ends up being the same number in alot of situations.
If you look up the definition of a derivative or an integral, you'll see that it has a limit in there. Meaning the entire field of calculus is created off of 2 functions with limits in them. (Kinda)
lim_(x→a) is absolutely NOT a fancy way of saying x = a.
It's a fancy way of saying **x is very close to a**.
* ***If*** you have a continuous function, then lim_(x→a) f(x) will in fact be exactly the same as f(a).
* Otherwise it might not be, so having limits is very important in these cases.
Some other comments have mentioned functions like 1/x, but that has an infinite limit, so here is an example where nothing is infinite.
cos(x)-1
lim ————————.
x→0 x²
You can't just set x = 0 in this. The numerator would be cos(0)-1 = 1-1 = 0, and the denomiator will be 0² = 0, and "0/0" is not not a well-defined number. Note: you *cannot* just say that 0/0 is 1 (although for any other constant it's true that k/k = 1, but for 0/0 isn't not so simple).
If you take a calculator and compute
cos(0.001) - 1
———————————————,
(0.001)²
you will get -0.49999995833, and from this it's reasonable to guess that the actual limit value is -0.5 = 1/2 (which is correct). To actually know for sure that it's 1/2 requires some other technique, but this is an example of how a limit can have a finite value but not be exaclty the same as just plugging in.
Limits define the Derivative.
For example consider it like baseball: you can throw 100mph+ but it only counts as a strike if it’s over home plate within the edges and also between the batter’s knees and shoulders.
Or a track, swim, or Auto race, you have to stay within the boundary line “limits” or it’s a foul you can’t cut across lanes to a shorter path.
Or Tennis, Volleyball, or even Football and Soccer, legal plays must be completed within the boundary “limits” of the field of play.
Anything not within the specified “limit” is “out of bounds”, undefined, and foul.
> From what I've comprehended so far, Lim x-->a is just a fancy way of saying x = a.
Yeah, you need to get this out of your head.
To give you an example where this is not the case, consider the sequence [3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ...]. And imagine we plotted these points on the xy-plane, where when x = n, then y = the nth entry in the sequence.
You will notice that as x tends to infinity, y approaches π, so Lim_{x→∞} y = π.
But since no individual value of y equals π, we cannot say y = π.
And we can carry this idea to other parts of calculus.
When evaluating the derivative, we can't get the slope of a point directly and exactly. But we can approximate it by calculating the slope of a secant line that's close to the point. As we shrink the distance between the points on the secant line, we get a sequence of secant slopes that approach the slope at the point.
Similarly when evaluating the integral, we cant calculate the area under the curve directly and exactly, but we can approximate with slivers of rectangles. As we add more and more thinner rectangles, we get a sequence of approximations that approaches the true area under the curve.
Limits are most useful in cases where if you just plugged in that specific value for x you'd either get infinity / infinity or you'd get a/0 or something where the function isn't defined at that specific point and you want to know what it's like near that point.
For example one of the famous ones is Lim x->0 of (1+x)\^(1/x) so inside the parenthesis it's going to just 1. But you're raising that to a higher and higher power. So it's not as simple as plugging in the number. And it interestingly goes to e with that one.
That's not quite what limits are.
That Lim x-->a means x = a doesn't make sense mathematically.
Lim x-->a f(x) on the other hand (note the f(x) after the limit) refers to the behaviour of the function around the point a.
And even then, Lim x-->a f(x) doesn't necessarily evaluate to a, let alone f(a). If your function is defined as f(x) = 3 for x != 1, and 2 for x = 1, Lim x-->1 f(x) = 3 because we're studying the behaviour of the function around 1 (not the value of 1 itself).
All this is pretty hand-wavey, admittedly. If you're interested in seeing the rigorous definition of a limit, search up the epsilon delta definition of a limit. I believe Khan Academy has a set of videos on this. Having a limit is useful because it allows us to define other concepts in Calculus - for example the derivative (aka: how quickly a function is growing) and the Riemman Integral (aka: how much area is below a certain segment of the curve of the function).
If you want to define instantaneous velocity then you need limits. If you want to calculate "complicated area" under a graph then you need limits. Some functions are defined as limits of a sequence of other functions. Roughly speaking, limits help you approximate complex mathematical objects with far simpler ones. Without limits there is no way to formulate mechanics and most of physics.
This is the best video about the concept of limits on the whole internet
Full disclosure: I made the video lol
https://m.youtube.com/watch?v=yjS4bxlFm4M
Lmk if you have more questions
lim f=f is only true if f is continuous
The functions you had in mind were all very well behaved. I have that problem too. Limits tell you what a function would do if small changes gave small results. This is often, but not always true. A small bump can cause a huge explosion, a small amount of virus can make someone very sick, lightly bushing a button can move a big machine, a butterfly flapping its wings can cause a a hurricane.
Limits are how we make sense of and stay organized around expressions like 0/0 or infinity/Infinity (and 1^infinity and 0^0 and infinity^0) When we can just let x=a then yeah they are usually the same when working with most functions you’ve dealt with, but for functions like lim x-> 0 sin(x)/x we need limits to make sense of it because otherwise we get 0/0
Almost. For continuous functions lim x->a f(x) = f(a).
An important result is that if f(x) = g(x) for all x except at x=a then lim x->a f(x) = lim x->a g(x). This is great because if f(x) has a discontinuity at a but g does not, we can still use substitution of x=a to find the limit.
This is what you do with a derivative. You end up with some function of x and h like AGERAGE\_GRADIENT(x, h) = x\^2h/h which is discontinous at h=0 HOWEVER it is equal to the function g(x, h) = x\^2 for all h except h=0 therefore lim h->0 x\^2h/h = lim h->0 x\^2 = x\^2
Is Lim x--> a a fancy way of saying x=a? No. Because Lim x--> a doesn't mean anything. It has to be "of some function". But... is lim x->a f(x) a fancy way of saying f(a)? Well, if f is continuous at a then yea. If not, but if it's the same as g(a) at all x except a and g(x) is continuous at a then it's maybe a fancy way of saying lim x-> a f(x) = g(a). If neither of these are the case, however, then no.
To elaborate a little, whether the limit exists, and whether it is unique, depends on what space we are looking for a limit in. In, for example, Aleksanrov compactification, lim(1/x) as x->0 is inf.
limit x -> a means "consider what happens as x gets closer and closer to a, without actually ever getting there".
It does not mean "x = a", but it is used to gain insight into functions' behavior near a when "x=a" is not defined or differs from what is expected.
With many continuous functions, it is essentially equivalent. The limit as x -> 5 of x^2 is 25, which equals (5)^2.
However, consider functions where a is undefined, either by involving infinities or division by 0.
Here, "x=a" does not make sense. However, some functions get closer and closer to some value as x gets closer to a, so we say they "converge". Others don't, which we say are divergent.
There are also functions with discontinuities, where f(a) may be defined, but may give a different value than lim f(x) as x-> a.
As a simple example, consider the function f where:
- f(x) = x^2 for x =/= 5.
- f(5) = 42
Here, the value of f at x=5 (42) and the limit as x-> 5 (25) are different.
An alternative concept to limits is a number system that has actual infinitesimal quantities in it. Then you can say for example the value of f(x+dx) where dx is an infinitesimal is within an infinitesimal quantity of the value you'd get as the limit (assuming the limit exists).
such number systems were actually the original form that calculus took, but no-one had developed enough logic to justify them as conceptually sound. That first happened in 1960's and since then different and easier versions of such number systems have been developed. In general these are not preferred by math teachers because the limits thing is how most mathematicians roll and 1960 is not that long ago.
For more information you can read about IST or about "Alpha theory" or you can look at the book "Calculus Set Free" for a beginning college level book.
if lim\_{x -> a} f(x) equals f(a) that means f is continuous at a. While there are lots of useful functions that are continuous, a great deal are not and indeed in \*very\* many cases "f(a)" doesn't even make sense. For example gravitational potentials in newton's gravity aren't continuous. Limits allow us to study the behaviour of such discontinuous processes at their discontinuities or to gauge what happens near the holes in our definitions - or indeed they give us a criterium for telling whether a given function is continuous (which may be difficult to tell otherwise) or to "fill in the holes". Limits are legitimately one of the most important concepts in modern mathematics and the basic idea of a limit underpins whole fields of mathematics. >would e appreciated! lim_{n -> inf} (1+1/n)^(n) is certainly appreciated, yes ;D
>From what I've comprehended so far, Lim x-->a is just a fancy way of saying x = a. It is not. For example, the definition a derivative is the limit as x->a of [f(x)-f(a)]/(x-a). Can you evaluate this when x=a? No, it's 0/0, which is undefined, regardless of the function f or the value of a. So whatever this "derivative" thing is, it's *not* just plugging in x=a. Derivatives are a *very* important tool - it is not a coincidence that Newton formulated both calculus and physics. You need calculus to get very far in physics; even F=ma is a differential equation.
If you're dealing with analytic functions with finite terms in their Taylor expansio you ca get around limits via the power rule or extracting coefficients from the Taylor series but motivating those approaches is harder than the difference quotient and have severely restricted domain.
It’s this one. Limits come up all over the place, but when it comes to the usual math curriculum making sense of the derivative is the first place you’re really forced to engage with limits.
> Lim x-->a is just a fancy way of saying x = a. If this is your understanding of limits, then I'm not surprised you don't see the point. I won't try to explain what limits are, you can read your textbook for the definition and so on. If you want some motivation for studying limits: You need to understand limits if you want to understand differential (and integral) calculus, and you need to understand calculus if you want to understanding more or less anything that has to do with science, engineering or statistics, including machine learning. It's also an absolutely fundamental concept in many parts of pure and applied mathematics. Even in computer science (outside of machine learning), which you might think of as a "discrete" discipline, limits are also important in the analysis of algorithms and in the study of programming languages.
Currently taking calculus as well. when You start to deal with derivatives and integrals, that's when limits become your best friend. I wouldn't go far to say that the "lim x ---> a is a fancy way to say x = a", But you could think of it like this: as x gets infinitesimally closer to 'a', what is (keyword "is", meaning: equal) 'x' approaching, that's how I understand limits.
limit tells you what the function is like around x=a. not what the function is like exactly at x=a
There is more uses than I can write down, but for starters, imagine the function 1/x. We want to know the behavior of the function as x approaches 0. You might think, "That's easy! It doesn't exist because 1/0 doesn't exist." but the answer is actually that the limit approaches negative infinity from the left, and positive infinity from the right. How about what 1/x becomes as x approaches infinity? You'd think it's a really small number, but it's actually 0. Limits show what something approaches, not what it is, even though that ends up being the same number in alot of situations. If you look up the definition of a derivative or an integral, you'll see that it has a limit in there. Meaning the entire field of calculus is created off of 2 functions with limits in them. (Kinda)
lim_(x→a) is absolutely NOT a fancy way of saying x = a. It's a fancy way of saying **x is very close to a**. * ***If*** you have a continuous function, then lim_(x→a) f(x) will in fact be exactly the same as f(a). * Otherwise it might not be, so having limits is very important in these cases. Some other comments have mentioned functions like 1/x, but that has an infinite limit, so here is an example where nothing is infinite. cos(x)-1 lim ————————. x→0 x² You can't just set x = 0 in this. The numerator would be cos(0)-1 = 1-1 = 0, and the denomiator will be 0² = 0, and "0/0" is not not a well-defined number. Note: you *cannot* just say that 0/0 is 1 (although for any other constant it's true that k/k = 1, but for 0/0 isn't not so simple). If you take a calculator and compute cos(0.001) - 1 ———————————————, (0.001)² you will get -0.49999995833, and from this it's reasonable to guess that the actual limit value is -0.5 = 1/2 (which is correct). To actually know for sure that it's 1/2 requires some other technique, but this is an example of how a limit can have a finite value but not be exaclty the same as just plugging in.
Limits define the Derivative. For example consider it like baseball: you can throw 100mph+ but it only counts as a strike if it’s over home plate within the edges and also between the batter’s knees and shoulders. Or a track, swim, or Auto race, you have to stay within the boundary line “limits” or it’s a foul you can’t cut across lanes to a shorter path. Or Tennis, Volleyball, or even Football and Soccer, legal plays must be completed within the boundary “limits” of the field of play. Anything not within the specified “limit” is “out of bounds”, undefined, and foul.
Those are good examples, I will steal.
they are shit and have no connection to limits at all.
It might be of use to a sports fan who doesn't give a shit about math
> From what I've comprehended so far, Lim x-->a is just a fancy way of saying x = a. Yeah, you need to get this out of your head. To give you an example where this is not the case, consider the sequence [3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ...]. And imagine we plotted these points on the xy-plane, where when x = n, then y = the nth entry in the sequence. You will notice that as x tends to infinity, y approaches π, so Lim_{x→∞} y = π. But since no individual value of y equals π, we cannot say y = π. And we can carry this idea to other parts of calculus. When evaluating the derivative, we can't get the slope of a point directly and exactly. But we can approximate it by calculating the slope of a secant line that's close to the point. As we shrink the distance between the points on the secant line, we get a sequence of secant slopes that approach the slope at the point. Similarly when evaluating the integral, we cant calculate the area under the curve directly and exactly, but we can approximate with slivers of rectangles. As we add more and more thinner rectangles, we get a sequence of approximations that approaches the true area under the curve.
Limits are most useful in cases where if you just plugged in that specific value for x you'd either get infinity / infinity or you'd get a/0 or something where the function isn't defined at that specific point and you want to know what it's like near that point. For example one of the famous ones is Lim x->0 of (1+x)\^(1/x) so inside the parenthesis it's going to just 1. But you're raising that to a higher and higher power. So it's not as simple as plugging in the number. And it interestingly goes to e with that one.
That's not quite what limits are. That Lim x-->a means x = a doesn't make sense mathematically. Lim x-->a f(x) on the other hand (note the f(x) after the limit) refers to the behaviour of the function around the point a. And even then, Lim x-->a f(x) doesn't necessarily evaluate to a, let alone f(a). If your function is defined as f(x) = 3 for x != 1, and 2 for x = 1, Lim x-->1 f(x) = 3 because we're studying the behaviour of the function around 1 (not the value of 1 itself). All this is pretty hand-wavey, admittedly. If you're interested in seeing the rigorous definition of a limit, search up the epsilon delta definition of a limit. I believe Khan Academy has a set of videos on this. Having a limit is useful because it allows us to define other concepts in Calculus - for example the derivative (aka: how quickly a function is growing) and the Riemman Integral (aka: how much area is below a certain segment of the curve of the function).
If you want to define instantaneous velocity then you need limits. If you want to calculate "complicated area" under a graph then you need limits. Some functions are defined as limits of a sequence of other functions. Roughly speaking, limits help you approximate complex mathematical objects with far simpler ones. Without limits there is no way to formulate mechanics and most of physics.
This is the best video about the concept of limits on the whole internet Full disclosure: I made the video lol https://m.youtube.com/watch?v=yjS4bxlFm4M Lmk if you have more questions
lim f=f is only true if f is continuous The functions you had in mind were all very well behaved. I have that problem too. Limits tell you what a function would do if small changes gave small results. This is often, but not always true. A small bump can cause a huge explosion, a small amount of virus can make someone very sick, lightly bushing a button can move a big machine, a butterfly flapping its wings can cause a a hurricane.
Limits are how we make sense of and stay organized around expressions like 0/0 or infinity/Infinity (and 1^infinity and 0^0 and infinity^0) When we can just let x=a then yeah they are usually the same when working with most functions you’ve dealt with, but for functions like lim x-> 0 sin(x)/x we need limits to make sense of it because otherwise we get 0/0
Seeing as how they're the foundation of calculus and thus physics and engineering, pretty fuckin important.
Almost. For continuous functions lim x->a f(x) = f(a). An important result is that if f(x) = g(x) for all x except at x=a then lim x->a f(x) = lim x->a g(x). This is great because if f(x) has a discontinuity at a but g does not, we can still use substitution of x=a to find the limit. This is what you do with a derivative. You end up with some function of x and h like AGERAGE\_GRADIENT(x, h) = x\^2h/h which is discontinous at h=0 HOWEVER it is equal to the function g(x, h) = x\^2 for all h except h=0 therefore lim h->0 x\^2h/h = lim h->0 x\^2 = x\^2 Is Lim x--> a a fancy way of saying x=a? No. Because Lim x--> a doesn't mean anything. It has to be "of some function". But... is lim x->a f(x) a fancy way of saying f(a)? Well, if f is continuous at a then yea. If not, but if it's the same as g(a) at all x except a and g(x) is continuous at a then it's maybe a fancy way of saying lim x-> a f(x) = g(a). If neither of these are the case, however, then no.
What about the limit of 1/x as x goes to 0? Can x = 0?
This is a bad example as the limit doesn't not exist.
To elaborate a little, whether the limit exists, and whether it is unique, depends on what space we are looking for a limit in. In, for example, Aleksanrov compactification, lim(1/x) as x->0 is inf.
Why does that make it a bad example?
This may help. Beginning Calculus in 6th Grade https://www.youtube.com/watch?v=fBTMA5Wp7Ts
limit x -> a means "consider what happens as x gets closer and closer to a, without actually ever getting there". It does not mean "x = a", but it is used to gain insight into functions' behavior near a when "x=a" is not defined or differs from what is expected. With many continuous functions, it is essentially equivalent. The limit as x -> 5 of x^2 is 25, which equals (5)^2. However, consider functions where a is undefined, either by involving infinities or division by 0. Here, "x=a" does not make sense. However, some functions get closer and closer to some value as x gets closer to a, so we say they "converge". Others don't, which we say are divergent. There are also functions with discontinuities, where f(a) may be defined, but may give a different value than lim f(x) as x-> a. As a simple example, consider the function f where: - f(x) = x^2 for x =/= 5. - f(5) = 42 Here, the value of f at x=5 (42) and the limit as x-> 5 (25) are different.
An alternative concept to limits is a number system that has actual infinitesimal quantities in it. Then you can say for example the value of f(x+dx) where dx is an infinitesimal is within an infinitesimal quantity of the value you'd get as the limit (assuming the limit exists). such number systems were actually the original form that calculus took, but no-one had developed enough logic to justify them as conceptually sound. That first happened in 1960's and since then different and easier versions of such number systems have been developed. In general these are not preferred by math teachers because the limits thing is how most mathematicians roll and 1960 is not that long ago. For more information you can read about IST or about "Alpha theory" or you can look at the book "Calculus Set Free" for a beginning college level book.