## 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 $x - 2$.

## GeoGebra Advanced Tutorial 7 – Embedding Form Controls and JavaScript I

We have discussed before how to export a GeoGebra file into dynamic html. In all of our GeoGebra dynamic HTML documents, we have not allowed input from user except by dragging objects within the applet. In this tutorial, we are going to use HTML Form commands and basic Javascript to get input from the user using a text box and a command button.  In particular, we are going to display a dynamic HTML document with a Cartesian coordinate, a text box where the user will input the equation of the function that they want to graph, and a command button that will execute the graph command when clicked.  The final output of our tutorial can be viewed here and the html file (javascript1.html) can be downloaded  here.

If you have a background in HTML and Javascript, this tutorial will be quite easy; otherwise, read the  explanation about the HTML Form and Javascript commands below. » Read more

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