limits

bhaiyya can u give sum simple example like how to find the following limit by that method..
lim(x→2) 2x+3

This is useful when you have conjectured a limit (or its non existence) and want to prove that conjecture.

Here, the limit seems to be 7.

So, we have to prove that for any ε>0, we can exhibit δ(ε) such that for all x such that |x-2|<δ, it is true that |f(x) - 7 |<ε

This is best remembered as the ε-δ challenge. Someone gives you an ε which is usually very very small and you are expected to produce δ corresponding to this ε. That is why we write δ(ε).

So now,

|f(x) - 7| < \epsilon \Rightarrow |2x+3-7| < \epsilon \Rightarrow |x-2| < \frac{\epsilon}{2}

So if we choose \delta = \frac{\epsilon}{2} the stipulated challenge is met.

Hence we have proved that the limit is indeed 7.

This method is then used to establish other methods such as the Sandwich Principle which are more useful to evaluate limits. For example \lim_{x \rightarrow 0} \frac{\sin x}{x}

no sir... i cudnt get anything ...because our whole book on limits is completely based on epsilon delta... theres no concept of LHL and RHL... so maybe i need to just forget wats written in the book and read that again.. with cool mind.. maybe i can understand then