Problem Solving and Proofs Archives - Page 7 of 9

Limit by epsilon-delta proof: Example 2

November 1, 2011 GB Calculus and Analysis, Problem Solving and Proofs

This is the overdelayed continuation of the discussion on the $\epsilon-\delta$ definition of limits. In this post, we discuss another example.

Prove that the $\lim_{x \to 2} x^2 = 4$ .

Recall that the definition states that the limit of $f(x) = L$ as $x$ approaches $a$ if for all $\epsilon > 0$ , however small, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$ , then $|f(x) - L| < \epsilon$ .

From the example 1, we have learned that we should manipulate $|f(x)-L=|x^2 - 4|$ , to make one of the expressions look like $|x-a|=|x-2|$ . Solving, we have

$|f(x) - L| = |x^2 - 4| = |(x+2)(x-2)| = |x+2||x-2|$ .

Note that we have accomplished our goal, going back to the definition, this means that if $0 < x - 2 < \delta$ , then $|x+2||x-2| < \epsilon$ .

Now, it is not possible to divide both sides by $x + 2$ (making it $|x-2| < \frac{\epsilon}{|x+2|})$ because $x$ varies. This means that we have to find a constant $k$ such that $|x + 2| < k$ . » Read more

More examples of proof by contradiction

July 1, 2011 GB High School Geometry, High School Mathematics, Problem Solving and Proofs

We have had good discussions on mathematical proofs, so I am planning to create a mathematical proof series that will discuss the basics such as direct proof, indirect proof, and proof by mathematical induction. But before I do that, let me continue with more examples of proof by contradiction.

Proof by contradiction, as we have discussed, is a proof strategy where you assume the opposite of a statement, and then find a contradiction somewhere in your proof. Finding a contradiction means that your assumption is false and therefore the statement is true. Below are several more examples of this proof strategy.

Example 1: $\sqrt{2}$ irrational.

Example 2: $\sqrt{6}$ is irrational. The proof of this is basically the same as example 1, so it is left as an exercise.

Example 3: Proof that there are infinitely many primes.

Example 4: Knights and Liars

Example 5: $\sqrt{2} + \sqrt{3}$ is irrational. » Read more

3 comments irrationality of square root of 2, mathematical proofs, proof by contradiction, proof of the infinitude of primes, proof strategies

Simon’s Favorite Factoring Trick

June 1, 2011 GB High School Algebra, Problem Solving and Proofs

Hmmm... I didn't know Simon was that good in math.

When I was quite younger, one of my hobbies was joining internet forums (fora?) on problem solving. I was not really good at it, so my role was only to ask questions. One of the internet forums I joined was the Art of Problem Solving math forum.

Art of Problem Solving (AOPS) is a community of problem solvers dedicated for math competitions – probably the best place on the web to ask hard (and very hard) math questions. One of the tricks I learned there was Simon’s Favorite Factoring Trick (SFFT), a factorization technique popularized by one AOPS member. The general strategy (see example 3) of SFFT is to add a constant or variable to an expression to make it factorable. This strategy can also be named as “completing rectangle” in analogy with “completing the square.” » Read more

Leave a comment art of problem solving, completing the rectangle, completing the square, math competition books, math competitions, math contest books, math contest problems, math forum, online math help, problem solving books, simon's favorite factoring trick

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