epsilon-delta problems Archives - Math and Multimedia

Limit by epsilon-delta proof: Example 1

February 27, 2011 GB Calculus and Analysis, College Mathematics

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 $x - 2$ .