- #1
Alpharup
- 225
- 17
Well've I recently passed my school and am entering college...In my school, I have been exposed to intuitive calculus...ie...I learned only the basic idea of limits, continuity ad differentiability...Then I proceeded to integral calculus, applications of calculus, differential equations, etc...After learning all these, I found that I was in a level for learning the exact definition of limit, continuity, etc...
Now, I browsed through a lot of websites for learning the epsilon-delta definition of limit..Though I can get an idea that the intuitive treatment of limit exactly equals the formal definition, I'am not satisfied..If I look in one angle, I find that I understood the definition..If I approach in another angle, I find that I do not understand anything about it...This is because I want to prove that a certain limit is wrong and Iam not able to prove by this definition...
For example...
we have to prove
lim 2x+1=5...
x→2
I will give the method of solving this problem...Please point me if my method is wrong..
The actual definition of limit is
Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. Then the statement
lim f(x)=L
x→c
means
for all real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − c | < δ, we have |f(x) − L| < ε ( From Wikipedia)
1. By the definition of limit, I assume that there exists a δ > 0 such that for all x with 0 < |x − c | < δ..
2. Then I also assume that |f(x) − L| < ε is true..
3. Then by using these assumptions, I prove that ε exists and is >0...which is accoardance with the above definition..
4.Since all the conditions satisfy, the limit is true...
Now, I give this contradictiory statement
lim 2x+1= 5.1
x→2
How do you disprove this limit by the formal definition? If a certain statement which is true can be proved by a definition , then a certain statement which is false can also be disproved by the same definition...
I just want to disprove the statement, please help me out with this concept...
Now, I browsed through a lot of websites for learning the epsilon-delta definition of limit..Though I can get an idea that the intuitive treatment of limit exactly equals the formal definition, I'am not satisfied..If I look in one angle, I find that I understood the definition..If I approach in another angle, I find that I do not understand anything about it...This is because I want to prove that a certain limit is wrong and Iam not able to prove by this definition...
For example...
we have to prove
lim 2x+1=5...
x→2
I will give the method of solving this problem...Please point me if my method is wrong..
The actual definition of limit is
Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. Then the statement
lim f(x)=L
x→c
means
for all real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − c | < δ, we have |f(x) − L| < ε ( From Wikipedia)
1. By the definition of limit, I assume that there exists a δ > 0 such that for all x with 0 < |x − c | < δ..
2. Then I also assume that |f(x) − L| < ε is true..
3. Then by using these assumptions, I prove that ε exists and is >0...which is accoardance with the above definition..
4.Since all the conditions satisfy, the limit is true...
Now, I give this contradictiory statement
lim 2x+1= 5.1
x→2
How do you disprove this limit by the formal definition? If a certain statement which is true can be proved by a definition , then a certain statement which is false can also be disproved by the same definition...
I just want to disprove the statement, please help me out with this concept...