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

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In summary, the conversation involved proving the limit of a function to be 12. The attempt at a solution involved using the definition of a limit and factoring the function to help determine the value of delta. The final answer was found to be delta = min(1, epsilon/10).
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nietzsche
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Homework Statement



Prove that

[tex]
\begin{equation*}
\lim_{x \to 2} x^2 + 5x -2 = 12
\end{equation*}
[/tex]

Homework Equations





The Attempt at a Solution



We want to prove that given [tex]\varepsilon > 0[/tex], there exists a [tex]\delta[/tex] such that

[tex]
0<|x-2|<\delta \Rightarrow |f(x) - 12| < \varepsilon
[/tex]

[tex]
\begin{equation*}
f(x)-12\\
= x^2+5x-2-12\\
= (x+7)(x-2)
\end{equation*}
[/tex]

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!
 
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  • #2
write x+7 as (x-2)+9
 
  • #3
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.
 
  • #4
can you use the fact that the the limit distributes of addition and products?
 
  • #5
i'm sorry I'm still confused. i have no idea where to go.
 
  • #6
never mind, i think i figured it out, but not with factoring it like that.

i got [tex]
\delta = min(1,\frac{\varepsilon}{10})
[/tex]
 

1. What is an epsilon-delta proof?

An epsilon-delta proof is a mathematical method used to formally prove that a limit of a function exists. It involves using two variables, epsilon and delta, to show that for any given input value, there exists a corresponding output value within a certain distance from the limit.

2. Why is it important to use an epsilon-delta proof?

Epsilon-delta proofs are important because they provide a rigorous and precise way to prove the existence of a limit. They are essential in many areas of mathematics, such as calculus and analysis, and are also used in engineering and physics to ensure accuracy in calculations.

3. How does a simpler epsilon-delta proof differ from a regular one?

A simpler epsilon-delta proof uses a more straightforward and intuitive approach to proving the existence of a limit. It often involves breaking down the proof into smaller steps and using simpler algebraic techniques, making it easier to understand and follow.

4. Can you give an example of a simpler epsilon-delta proof?

Sure, let's say we want to prove that the limit of the function f(x) = 2x + 1 as x approaches 3 is 7. We can start by assuming an arbitrary value for epsilon, let's say epsilon = 0.5. Then, we can find a corresponding delta value by rearranging the expression for the limit: delta = (epsilon - 1)/2. So, if we choose any x value within a distance of delta from 3, the output of the function will be within 0.5 units from the limit of 7. This proves the existence of the limit.

5. Are there any tips for writing a successful simpler epsilon-delta proof?

Some tips for writing a successful simpler epsilon-delta proof include clearly stating the limit you are trying to prove, showing all steps and calculations, and using simple and intuitive algebraic techniques. It is also helpful to break the proof down into smaller steps and to carefully choose values for epsilon and delta to ensure the proof is valid.

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