Solving Limits: Proving x→7 Sqrt(16-x)=3

  • Thread starter bezgin
  • Start date
  • Tags
    Limits
In summary, the conversation discusses the definition of limits and how to prove the limit of a function using the delta-epsilon relation. It also touches on the concept of removable discontinuity and the use of analytic functions and residue theorem in approaching a value from different directions. The main point is that the limit of a function represents the value of the function at a specific point, regardless of the direction of approach.
  • #1
bezgin
22
0
I have a serious problem with understanding the definition of limits.

Prove that Lim(x->7) Sqrt(16-x)=3

I'd be grateful if you could explain why you do each step when you solve this question. Thanks.
 
Mathematics news on Phys.org
  • #3
Another question:

If, say, Lim(x->a) f(x) = infinity (when you approach from both sides), we call the point x=a as a removable discontinuity. Why? How can we remove it? If the limit approached to a value such as c, then we could define the function to be f(a) = c
Now, when it approaches infinity, we still call it removable discontinuity but it can't be removed by assigning a value!
 
  • #4
Sirus said:
http://en.wikipedia.org/wiki/Limit_of_a_function

By substituting x=7 into [itex]f(x)=\sqrt{16-x}[/itex], we find that when x approaches 7, f(x) approaches 3.

I meant to prove it by using the delta-epsilon relation. Substitution doesn't prove anything, of course. But I don't really understand HOW the delta-epsilon relation does.
 
  • #5
Please reply! I have a midterm on saturday.
 
  • #6
Many other PF members are much better qualified to answer this than I am. AFAIK, delta-epsilon relations are used to define continuity, not really to explain limits.
 
  • #7
bezgin said:
If, say, Lim(x->a) f(x) = infinity (when you approach from both sides), we call the point x=a as a removable discontinuity.

Who called such a discontinuity removable? You might want to check your definitions carefully. If a one sided limit "equals" infinity most definitions will say the limit does not exist (this isn't "equal" in the usual sense, it's really a way of keeping track of how the limit diverges). This goes for two sided limits as well-even if the left and right handed limits are both infinity, most definitions will say the limit does not exist.


Sirus, epsilon-delta's are very much a part of the rigorous definition of limits.
 
  • #8
First you need to know the exact interpretation if LIMIT CONCEPT.Perhaps a knowledge of Analytic function and residue theorem can help you.
Take for instance,a point in a number line can be approached from different directions(ideally infinity),ie,through X axis or through y-axis or even in an oblique axis.Limit of a function accentuates upon the point that no matter whatever direction we take to approach a value,the vlaue of the fuction at that partiicular point is the result that you get(in your case it is 3).This is what limit of a function denotes.Thats why when we take Z transform we rely upon jordn contour and the region of convergence is taken as the distance between two poles along the path of traversal.

Regards
drdolittle
 
  • #9
is it too late to be of help? i know your test is over but is there another one later? the answers so far are not much to the point.
 

1. What is a limit?

A limit is the value that a function approaches as the input (x) approaches a certain value. In other words, it is the value that the function is getting closer and closer to as x gets closer to a specific number.

2. How do you prove a limit?

To prove a limit, you need to show that as x approaches the given value, the function also approaches the given value. This can be done through various methods such as substitution, algebraic manipulation, and epsilon-delta proofs.

3. What is the process for solving a limit?

The process for solving a limit involves plugging in the given value for x and simplifying the function until you can determine the limit. This may involve factoring, rationalizing, or using other algebraic techniques.

4. How do you prove x→7 Sqrt(16-x)=3?

To prove this limit, we can use substitution and algebraic manipulation. First, we can substitute 7 for x in the function and simplify to get the value of 3. This shows that as x approaches 7, the function also approaches 3, proving the limit.

5. Can limits exist at a point where the function is undefined?

No, limits cannot exist at points where the function is undefined. A limit can only exist if the function is defined and approaches a specific value as the input approaches a certain number. If the function is undefined at that point, then the limit does not exist.

Similar threads

  • General Math
Replies
5
Views
1K
  • General Math
Replies
3
Views
760
  • General Math
Replies
3
Views
2K
Replies
2
Views
1K
Replies
8
Views
2K
Replies
10
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
809
Replies
2
Views
1K
  • Precalculus Mathematics Homework Help
Replies
10
Views
803
Back
Top