Proving the existence of a limit in 2 different methods When proving the existence of a limit, I know about 2 methods. One is showing that the Right hand limit is equal to the left hand limit. Another one is using the epsilon delta definition of limit. Both are presented, but which one is used for what purpose? Is there a specific reason to prefer one over other? Don't they portray the same idea?
 A: For the most part, if the functions are well-behaved the two methods are equivalent. However, $\epsilon$-$\delta$ definition is definitely more rigor, and usually introduced in more advanced classes. The right-hand, left-hand approach is more intuitive.
To see why they are equivalent, note that in the $\epsilon$-$\delta$ defintion we are showing that:
For any given $\epsilon > 0$, $\exists \delta$ such that $\forall x$
$$0<|x-c|<\delta \Rightarrow |f(x)-L| < \epsilon$$
The right-hand, left-hand approach, in the language of $\epsilon$-$\delta$ definition, states that
$\forall \epsilon > 0, \exists \delta > 0 \text{ such that }$
$$0 < x - c < \delta \implies f(x) - L < \epsilon$$ and
$$0 < c - x < \delta \implies L - f(x) < \epsilon $$
This method is based on the idea that the function gets arbitrarily close to a particular value $L$ as the input gets arbitrarily close to a particular point $c$, and it is useful in situations where a more intuitive approach is needed.
In short, both methods are acceptable. Which one to use is a context-sensitive question. The right-hand, left-hand approach is more intuitive and might be an easier, but $\epsilon$-$\delta$ approach is rigorous and considered formal.
