/cry Limits

[Suicidesoldier#1]

Limits

Limits, as in mathematics. What the hell are they?! eek

I originally assumed that, you know, it was the limit to something. However, it seems that you can solve a problem normally without the limit even interfering- or being a major part of the problem.

What the hell does a limit do, then, if it's primary job isn't to limit or present a limit of something, and what is it's purpose in mathematics? O_o  

[SuchSweetSadism]

the value that a function or sequence "approaches" as the input or index approaches some value  

[jestingrabbit]

SuchSweetSadism

the value that a function or sequence "approaches" as the input or index approaches some value

Yeah, this is right.

So, say you had

limit as n to infinity [ 1/n ]

As you plug bigger and bigger values for n into 1/n you get numbers that are smaller and smaller, and closer and closer to 0. So

limit as n to infinity [ 1/n ] = 0

Understanding limits is really important when you go and try to understand calculus, so they're worth your time and effort.  

[saphria_eragon]

Limits are the building blocks in calculus. Without limits you can not understand derivatives and without derivatives you will not understand integration.

So just hang in there with limits.  

[Suicidesoldier#1]

So, if something was like, what is the limit of x as x approaches 2, in f(x)= 2x, what would I do?  

[VorpalNeko]

In practice, one learns to recognize which functions are continuous, i.e., have limits equal to their value, so that one can simply plug in the limit point into the function. Polynomials are always continuous, so lim_{x->a}[ f(x) ] = f(a) for any polynomial f(x).

Actually proving a function is continuous at a point is not that simple. The definition of continuity at x = a:
for all ε>0, there exists δ>0 such that |x-a|<δ implies |f(x)-f(a)|<ε
The intuitive idea is that "if you tell me how close to f(a) you want the function to be, I can always tell you how close x has to be to a in order to guarantee it."

So for your example of f(x) = 2x, one sees that for any given ε>0, one can just pick δ = ε/2 (or anything less than ε/2). Then: |x-a|<δ
⇒ a-δ < x < a+δ
⇒ 2a - 2δ < 2x < 2a+2δ
⇒ -ε < f(x) - 2a < ε
⇒ |f(x) - f(a)| < ε

Again, in practice one can prove very general theorems like:
(1) If f(x) is continuous, then cf(x) for any constant c.
(2) If f(x) and g(x) are continuous, then so is their sum.
(3) If f(x) and g(x) are continuous, then so is their product.
(4) If f(x) and g(x) are continuous, then so is their quotient whenever g(x) is nonzero.
The properties (1-3) directly imply that every polynomial is continuous from continuity of f(x) = x. And once you know that, you no longer have to play the ε-δ game: "aha, it's a polynomial, so it's limit is the same as its value."  

[viper232]

VorpalNeko

In practice, one learns to recognize which functions are continuous, i.e., have limits equal to their value, so that one can simply plug in the limit point into the function. Polynomials are always continuous, so lim_{x->a}[ f(x) ] = f(a) for any polynomial f(x).

Actually proving a function is continuous at a point is not that simple. The definition of continuity at x = a:
for all ε>0, there exists δ>0 such that |x-a|<δ implies |f(x)-f(a)|<ε
The intuitive idea is that "if you tell me how close to f(a) you want the function to be, I can always tell you how close x has to be to a in order to guarantee it."

So for your example of f(x) = 2x, one sees that for any given ε>0, one can just pick δ = ε/2 (or anything less than ε/2). Then: |x-a|<δ
⇒ a-δ < x < a+δ
⇒ 2a - 2δ < 2x < 2a+2δ
⇒ -ε < f(x) - 2a < ε
⇒ |f(x) - f(a)| < ε

Again, in practice one can prove very general theorems like:
(1) If f(x) is continuous, then cf(x) for any constant c.
(2) If f(x) and g(x) are continuous, then so is their sum.
(3) If f(x) and g(x) are continuous, then so is their product.
(4) If f(x) and g(x) are continuous, then so is their quotient whenever g(x) is nonzero.
The properties (1-3) directly imply that every polynomial is continuous from continuity of f(x) = x. And once you know that, you no longer have to play the ε-δ game: "aha, it's a polynomial, so it's limit is the same as its value."

crying

I wanted to provide a δ, ε definition but I'm not nearly as clear and concise as you are when writing out math.

Clearly you're a grad student who has had to deal with undergraduates.