Question: How to Justify the Steps in Proving a Limit?

In summary: Thanks!In summary, the conversation is about proving that the limit of a function approaches a specific value. The conversation includes a discussion on how to show that the function is less than a given value and the importance of using absolute value in the proof. The summary also mentions a side question about a specific step in the proof.
  • #1
hmm?
19
0
Hello,

I'm having trouble with a step;I was wondering if someone could shed some light on me. Thanks.

Prove that lim x->-2 (x^2-1)=3

0<|f(x)-L|<epsilon whenever 0<|x-a|<delta

0<|(x^2-1)-3|<epsilon whenever 0<|x-(-2)<delta
= |x^2-4|<epsilon
= |(x-2)(x+2)|<epsilon
= |x-2||x+2|<epsilon
= If |x-2|<C (C=constant) then |x-2||x+2|<C|x+2|
= C|x+2|<epsilon = |x+2|<epsilon/C

Assume |x+2|<1 so -1<x+2<1 = -3<x<-1 = -5<x-2<-3

After this step -5<x-2<-3, I'm inclined to set |x-2|<-3, but book states that |x-2|<5. I'm lost at this step, should it be|-5|? Anyways, any explanations would be great.

Thanks,
Chris
 
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  • #2
hmm? said:
Assume |x+2|<1 so -1<x+2<1 = -3<x<-1 = -5<x-2<-3

After this step -5<x-2<-3, I'm inclined to set |x-2|<-3, but book states that |x-2|<5. I'm lost at this step, should it be|-5|?

You've shown that: |x+2|<1 <=> -5<x-2<-3
This implies that |x-2|<5, simply because x-2<-3 means that surely x-2<5, so that -5<x-2<5, or |x-2|<5.

You wrote |x-2|<-3, which is never true becuase the left side is positive and the right side isn't. Did you mean |x-2|<3?
 
  • #3
Small detail that is important:

Prove that lim x->-2 (x^2-1)=3

0<|f(x)-L|<epsilon whenever 0<|x-a|<delta

You only want to prove that "|f(x)-L|<epsilon whenever 0<|x-a|<delta".

In other words, f(x) can be = L.
 
  • #4
Perhaps this way would work:

Prove that lim x->-2 (x^2-1)=3
let E > 0 be given
and:
|f(x) - L|
= |x^2 - 4|
= |(x - 2)(x + 2)|
= |x - 2||x + 2|
if 0 < |x + 2| < 1 then,
-3 < x < -1
|x - 2| < 5
< 5|x + 2|
therefore:
if 0 < |x + 2| < 1 and 5|x + 2| < E
then by transitivity of <(less than), |f(x) - L| < E
or reworded:
if D = minimum(1, E/5), then |f(x) - L| < E

(where E = epsilon; D = delta)
 
  • #5
Sorry this part:
if 0 < |x + 2| < 1 then,
-3 < x < -1
|x - 2| < 5
is a "side part"
 
  • #6
Galileo said:
You've shown that: |x+2|<1 <=> -5<x-2<-3
This implies that |x-2|<5, simply because x-2<-3 means that surely x-2<5, so that -5<x-2<5, or |x-2|<5.

Alright this kinda clarifies my question, so -5 is like the min? Which would grant -5<x-2<5--since epsilon can never be E<0 - the absolute value is necessary? Sorry if I come off a bit slow, but it's just that I'm trying to justify every step so I completely understand the concept.
 
Last edited:

1. What is a limit in mathematics?

A limit in mathematics is a fundamental concept in calculus that describes the behavior of a function as its input values approach a specific value. It is used to determine the value that a function approaches as the input values get closer and closer to a certain point.

2. How is a limit calculated?

A limit is calculated by evaluating the function at values that are increasingly closer to the specified point. These values are known as the limit's approach values. The limit is then determined by observing the behavior of the function as the approach values get closer and closer to the specified point.

3. What does it mean for a limit to exist?

A limit exists when the function has a well-defined output value as its input values approach a specific point. In other words, the function's behavior is consistent and does not have any abrupt changes or gaps as the input values get closer to the specified point.

4. What are the two types of limits?

The two types of limits are one-sided limits and two-sided limits. One-sided limits describe the behavior of a function as its input values approach a specific point from one side only (either from the left or the right). Two-sided limits, on the other hand, describe the behavior of a function as its input values approach a specific point from both sides.

5. How is the concept of a limit used in real-world applications?

The concept of a limit is used in many real-world applications, such as in physics, engineering, and economics. For example, it can be used to determine the maximum speed of a moving object, the stability of a structure, or the optimal production level for a company. In these applications, the limit represents a boundary or a threshold that must be considered for the system to function properly.

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