A question

[bezgin]

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.

[Sirus]

http://en.wikipedia.org/wiki/Limit_of_a_function

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

[bezgin]

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!

[bezgin]

http://en.wikipedia.org/wiki/Limit_of_a_function

By substituting x=7 into f(x)=\sqrt{16-x}, 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.

[bezgin]

Please reply! I have a midterm on saturday.

[Sirus]

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.

[shmoe]

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.

[drdolittle]

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

[mathwonk]

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.