Well've I recently passed my school and am entering college...In my school, I have been exposed to intuitive calculus...ie...I learnt 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...(adsbygoogle = window.adsbygoogle || []).push({});

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....

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# How to disprove this limit?

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