If you want to learn Calculus, the websites below will most likely help to you. Most of these websites contain conceptual and intuitive explanations of Calculus concepts and most of them are interactive.
All in all, they contain more than 22,000 calculus tutorials and computations. » Read more
This is the overdelayed continuation of the discussion on the definition of limits. In this post, we discuss another example.
Prove that the .
Recall that the definition states that the limit of as approaches if for all , however small, there exists a such that if , then .
From the example 1, we have learned that we should manipulate , to make one of the expressions look like . Solving, we have
Note that we have accomplished our goal, going back to the definition, this means that if , then .
Now, it is not possible to divide both sides by (making it because varies. This means that we have to find a constant such that . » Read more
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 .
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