Limits and Continuity - Absolute Value Technicality ....

In summary, a limit is a mathematical concept that determines the behavior of a function as its input values approach a certain point. The absolute value technicality is used in evaluating limits with discontinuities, while continuity refers to the smoothness of a function and differentiability refers to the existence of a derivative. The intermediate value theorem can be used to determine continuity, and common techniques used to evaluate limits include direct substitution, factoring, algebraic manipulation, and trigonometric identities, among others.
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I am reading Manfred Stoll's Book: "Introduction to Real Analysis" ... and am currently focused on Chapteer 4: Limits and Continuity ...

I need some help with an inequality involving absolute values in Example 4.1.2 (a) ... Example 4.1.2 (a) ... reads as follows:View attachment 7247In the above text we read ...

"... If \(\displaystyle \mid x - p \lvert \ \lt \ 1\) then \(\displaystyle \mid x \mid \ \lt \ \mid p \mid \ + \ 1\) ... "Can someone please show me how to rigorously prove the above statement ...

Peter
 
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Peter said:
"... If \(\displaystyle \mid x - p \lvert \ \lt \ 1\) then \(\displaystyle \mid x \mid \ \lt \ \mid p \mid \ + \ 1\) ... "Can someone please show me how to rigorously prove the above statement ...
This comes from the triangle inequality $|a+b|\leqslant |a| + |b|$. With $a=x-p$ and $b=p$, that becomes $$|x| = |(x-p) + p| \leqslant |x-p| + |p| <1 + |p|.$$
 
  • #3
Opalg said:
This comes from the triangle inequality $|a+b|\leqslant |a| + |b|$. With $a=x-p$ and $b=p$, that becomes $$|x| = |(x-p) + p| \leqslant |x-p| + |p| <1 + |p|.$$
Thanks Opalg ... obvious when you see it ... :( ...

Appreciate your help ...

Peter
 
  • #4
Peter said:
Thanks Opalg ... obvious when you see it ... :( ...
One of those little tricks that eventually become second nature. :)
 

Related to Limits and Continuity - Absolute Value Technicality ....

1. What is the definition of a limit?

A limit is a mathematical concept that describes the behavior of a function as its input values approach a certain value or point. It essentially determines what value a function is approaching at a specific point.

2. How is the absolute value technicality used in limits and continuity?

The absolute value technicality is used to evaluate limits where the function has a discontinuity at a specific point. It involves taking the limit from the left and right sides of the point and comparing the two values to determine if the limit exists.

3. What is the difference between continuity and differentiability?

Continuity refers to the smoothness and unbrokenness of a function, where there are no sudden jumps or breaks in the graph. Differentiability, on the other hand, refers to the existence of a derivative at a specific point.

4. How can you use the intermediate value theorem to determine continuity?

The intermediate value theorem states that if a function is continuous on a closed interval, then it takes on every value between the smallest and largest output values. Therefore, if a function satisfies this condition, it is considered continuous on that interval.

5. What are the common techniques used to evaluate limits?

The most common techniques used to evaluate limits include direct substitution, factoring and simplifying, using algebraic manipulation, and using trigonometric identities. Other techniques such as L'Hôpital's rule and Taylor series expansions can also be used in more complex cases.

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