Delta/Epsilon Limit Proof

In summary, the limit of f(x) as x approaches -5 is -4. To find δ, you want to make sure that if x is within δ units of -5, then f(x) will be within 0.5 units of -4.
  • #1
iRaid
559
8

Homework Statement


Find [itex]L=\lim_{x\rightarrow\x_{0}} f(x)[/itex]. Then find a number [itex]\delta > 0[/itex] such that for all x, [itex]0<\left|x-x_{0}\right|<\delta[/itex] [itex]\Rightarrow[/itex] [itex]\left|f(x) - L\right|<\epsilon[/itex]

Problem:
[itex]f(x)=\frac{x^{2}+6x+5}{x+5}[/itex], [itex]x_{0}=-5[/itex], [itex]\epsilon=0.5[/itex]

Homework Equations


The Attempt at a Solution


Found the limit first which = -4
[itex]\left|f(x) - L\right|<\epsilon[/itex]
[itex]\left|\frac{x^{2}+6x+5}{x+5} - 4\right|<\epsilon[/itex] <--- Problem here not sure... My teacher seems to sometimes keep the negative limit or sometimes he'll make it positive :confused:
[itex]\left|\frac{(x+5)(x+1)}{x+5} - 4\right|<.05[/itex]
[itex]\left|x+1-4\right|<.05[/itex]
[itex]\left|x-3\right|<.05[/itex]
[itex]-.05<x-3<.05[/itex]
[itex]7.95<x+5<8.05[/itex]
[itex]\delta=.05[/itex]

Is that right :|PS: can someone tell me how to fix the limit in latex?

Thanks.
 
Last edited:
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  • #2
iRaid said:

Homework Statement


Find [itex]L=\lim_{x\rightarrow x_{0}} f(x)[/itex]. Then find a number [itex]\delta > 0[/itex] such that for all x, [itex]0<\left|x-x_{0}\right|<\delta[/itex] [itex]\Rightarrow[/itex] [itex]\left|f(x) - L\right|<\epsilon[/itex]

Problem:
[itex]f(x)=\frac{x^{2}+6x+5}{x+5}[/itex], [itex]x_{0}=5[/itex], [itex]\epsilon=0.5[/itex]

Homework Equations





The Attempt at a Solution


Found the limit first which = -4
[itex]\left|f(x) - L\right|<\epsilon[/itex]
[itex]\left|\frac{x^{2}+6x+5}{x+5} - 4\right|<\epsilon[/itex] <--- Problem here not sure... My teacher seems to sometimes keep the negative limit or sometimes he'll make it positive :confused:
[itex]\left|\frac{(x+5)(x+1)}{x+5} - 4\right|<.05[/itex]
[itex]\left|x+1-4\right|<.05[/itex]
[itex]\left|x-3\right|<.05[/itex]
[itex]-.05<x-3<.05[/itex]
[itex]7.95<x+5<8.05[/itex]
[itex]\delta=.05[/itex]

Is that right :|


PS: can someone tell me how to fix the limit in latex?

Thanks.
Removed an extra \.

First of all, your function is defined everywhere except at x = -5. For any other value of x, f(x) = x + 1. For this reason, [itex]\lim_{x \to 5} f(x) = 6[/itex], not -4.
 
  • #3
oops, x0 is supposed to equal -5 not 5.

Also, I only have 1 \ :|
 
  • #4
This is what you had:
[noparse] [itex]L=\lim_{x\rightarrow\x_{0}} f(x)[/itex] [/noparse]

The \ after rightarrow and before x_{0} was causing the problem.

Your value of δ is not right.

Let's sum up, with errors corrected.
[tex]\lim_{x \to -5}f(x) = -4[/tex]

For x [itex]\neq[/itex] -5, f(x) = [(x + 5)(x + 1)]/(x + 5)] = x + 1

You want to find δ so that if |x - (-5)| < δ, then |f(x) - (-4)| < 0.5.

Can you take it from here?
 
Last edited:
  • #5
-5+1=-4 not 6 tho >.<
 
  • #6
iRaid said:
-5+1=-4 not 6 tho >.<
Right. My previous post has been corrected.
 

1. What is a Delta/Epsilon Limit Proof?

A Delta/Epsilon Limit Proof is a method used in calculus to formally prove the limit of a function. It uses the concepts of delta and epsilon to show that as the input to a function approaches a particular value, the output of the function also approaches a specific value.

2. How does the Delta/Epsilon Limit Proof work?

The Delta/Epsilon Limit Proof starts by assuming a limit of a function exists at a particular point. Then, using the definition of a limit, it works backwards to find a relationship between the input and output of the function that satisfies the definition. This relationship is expressed in terms of delta and epsilon, representing the distance between the input and the limit and the distance between the output and the limit, respectively.

3. Why is the Delta/Epsilon Limit Proof important?

The Delta/Epsilon Limit Proof is important because it provides a rigorous and formal way to prove limits in calculus. It is used to show that a function approaches a specific value as the input approaches a particular point, which is essential for understanding the behavior of functions and their derivatives.

4. What is the role of delta and epsilon in a Delta/Epsilon Limit Proof?

In a Delta/Epsilon Limit Proof, delta and epsilon are used to represent the distance between the input and the limit, and the distance between the output and the limit, respectively. They are used to define a relationship between the input and output of a function that satisfies the definition of a limit.

5. Are there any limitations to the Delta/Epsilon Limit Proof?

While the Delta/Epsilon Limit Proof is a powerful tool for proving limits in calculus, it does have its limitations. It can only be used for continuous functions, and it may not always be possible to find a relationship between the input and output of a function that satisfies the definition of a limit. Additionally, it can be a complex and time-consuming process, so it may not always be the most efficient method for finding limits.

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