We have discussed extensively the meaning of the definition. In this post, we are going to learn some strategies to prove limits of functions by definition. The meat of the proof is finding a suitable for all possible values.
Recall that the definition states that the limit of as approaches , if for all , however small, there exists a such that if , then .
Example 1: Let . Prove that
If we are going to study definition limit above, and apply it to the given function, we have , if for all , however small, there exists a such that if , then . We want to find the value of , in terms of ; therefore, we can manipulate one of the inequalities to the other’s form. In particular, we will manipulate to an expression such that the expression inside the absolute value sign will become .