Proving the Limit of x^2 + 5x - 2 as x Approaches 2 Using Epsilon-Delta Proof

[nietzsche]

Homework Statement

Prove that

<br /> \begin{equation*}<br /> \lim_{x \to 2} x^2 + 5x -2 = 12<br /> \end{equation*}<br />

Homework Equations

The Attempt at a Solution

We want to prove that given \varepsilon &gt; 0, there exists a \delta such that

<br /> 0&lt;|x-2|&lt;\delta \Rightarrow |f(x) - 12| &lt; \varepsilon<br />

<br /> \begin{equation*}<br /> f(x)-12\\<br /> = x^2+5x-2-12\\<br /> = (x+7)(x-2)<br /> \end{equation*}<br />

So I have an (x-2) term there in the epsilon part. I don't know how to apply that information so that I can choose a delta. Suggestions please!

 

[lurflurf]

write x+7 as (x-2)+9

 

[nietzsche]

sorry, i don't follow

so when i write x+7 as (x-2)+9 i get

f(x) - 12
= (x-2)^2 + 9(x-2)

and it looks like it might be useful, but i don't know how to use it.

 

[aPhilosopher]

can you use the fact that the the limit distributes of addition and products?

 

[nietzsche]

i'm sorry I'm still confused. i have no idea where to go.

 

[nietzsche]

never mind, i think i figured it out, but not with factoring it like that.

i got <br /> \delta = min(1,\frac{\varepsilon}{10})<br />