Limiting Proof: Showing lim (x^2+3x) = 10 x→2

In summary, the conversation discusses how to use the precise definition to show that the limit of the function (x^2+3x) is equal to 10 as x approaches 2. The conversation also mentions using the fact that x^2+3x-10 can be factored and setting an appropriate value for delta to satisfy the given equation.
  • #1
roadworx
21
0

Homework Statement



Use the precise definition to show
[tex]lim (x^2+3x) = 10[/tex]
x[tex]\rightarrow[/tex]2

The Attempt at a Solution



Let [tex]\epsilon[/tex] > 0

[tex]x^2 + 3x - 10 < \epsilon [/tex]

[tex](x-2)^2 = x^2 - 4x + 4 [/tex]

This doesn't equal the equation. Add 7x, -14

[tex]\left| x-2 \right| ^2 + 7 \left| x-2 \right| [/tex]

So far it's alright. Now I need to get a value for [tex]\delta[/tex]

[tex]\epsilon[/tex] = [tex]\delta^2 + 7 \delta[/tex]

Now I'm totally confused. Normally I've used simply [tex]\delta[/tex] expressions like [tex]\delta[/tex] = [tex]\epsilon[/tex]/2. What should I say my [tex]\delta[/tex] is equal to in this case, and why?

So [tex]\left| (x^2 + 3x) -10 \right|[/tex] < [tex]\delta^2 + 7 \delta[/tex]

Any help?
 
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  • #2
It would be better to use the fact that x2+ 3x- 10= (x- 2)(x+ 5) so
|x2+ 3x- 10|= |x-2||x+ 5| and you want that less than [itex]\epsilon[/itex]. That will be true if [itex]|x-2|< \epsilon/|x+5|[/itex] but you want a constant. If [itex]\gamma> |x+5|[/itex] then [itex]\epsilon/\gamma< \epsilon/|x+5|[/itex] so you can set [itex]\delta= \epsilon/\gamma[/itex]. To find an upper bound on |x+ 5| remember that you are taking x close to 2 anyway: say |x- 2|< 1 so -1< x-2< 1. How large can x+ 5 be?
 

1. What is the definition of a limit?

A limit is a mathematical concept that describes the behavior of a function as its input variable approaches a certain value. It represents the value that a function approaches as the input variable gets closer and closer to a specific value.

2. How do you prove a limit using the epsilon-delta definition?

To prove a limit using the epsilon-delta definition, you need to show that for every epsilon (a small positive number) there exists a corresponding delta (a small positive number) such that if the distance between the input variable and the limiting value is less than delta, then the distance between the output of the function and the limiting value is less than epsilon.

3. What is the limit of a polynomial function?

The limit of a polynomial function is the value that the function approaches as the input variable gets closer and closer to a specific value. For example, in the limit statement lim(x^2 + 3x) = 10 as x approaches 2, the limit is equal to 10.

4. How do you prove a limit using algebraic manipulation?

To prove a limit using algebraic manipulation, you can use algebraic properties and rules to simplify the expression and show that it is equal to the limiting value. In the case of lim(x^2 + 3x) = 10 as x approaches 2, you can plug in the value of 2 for x and show that the resulting expression is equal to 10.

5. What does it mean when a limit does not exist?

If a limit does not exist, it means that the function does not approach a specific value as the input variable gets closer and closer to a certain value. This could be due to a discontinuity or a jump in the function, or the function could approach different values from different directions. In the case of lim(x^2 + 3x) = 10 as x approaches 2, the limit exists and is equal to 10.

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