## Limit by epsilon-delta proof: Example 2

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

## Limit by epsilon-delta proof: Example 1

We have discussed extensively the meaning of the $\epsilon-\delta$ 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 $\delta$ for all possible $\epsilon$ values.

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 $|f(x) - L| < \epsilon$.

Example 1: Let $f(x) = 3x + 5$.  Prove that $\lim_{x \to 2} f(x) = 11$

If we are going to study definition limit above, and apply it to the given function, we have $\lim_{x \to 2} 3x + 5 = 11$, if for all $\epsilon > 0$, however small, there exists a $\delta > 0$ such that if  $0 < |x - 2| < \delta$, then $|3x + 5 - 11| < \epsilon$.  We want to find the value of $\delta$, in terms of $\epsilon$; therefore, we can manipulate one of the inequalities to the other’s form.  In particular, we will manipulate $|3x + 5 - 11| < \epsilon$ to an expression such that the expression inside the absolute value sign will become $x - 2$.

## THE NUMBER 1

Welcome to the Mathematics and Multimedia Blog Carnival Number 1. Before beginning, let us see what’s so special about 1 as a number.

is a homophone of Juan (dela Cruz), the person who represents the Filipino people.  He is just like Uncle Sam of the United States. Hmmm… just a thought, should we change Juan dela Cruz to Manny Pacquiao?

is the multiplicative identity. Any number multiplied by 1 is equal to that number.

is the only number that is not prime nor composite.

is the number in French playing cards intead of Ace. It is the international dialing code for the United States and Canada.

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is equal to 1/21 + 1/22 + 1/23 + 1/24 + 1/25 + …

is equal to sin2 (a) + cos2 (a)

is equal to | Fn x Fn+3 – Fn+1 x Fn+2 | (F = Fibonacci numbers)

One is said to be a lonely number.  If you are as lonely as the number one, maybe reading the blog submissions below will be of great help. » Read more

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