Limit of a constant

[Millesa]

lim x→3 4 = 4
Why there is a need for this example to exist if its numeric value is 4
I am not neglecting this situation, but why does it exist and where did it most likely come from?

 

[HallsofIvy]

Yes, that is the most trivial limit problem there is! The problem "exits" pretty much to see if you were paying attention! It is also useful for more complicated problems such as "limit as x goes to a of \(\displaystyle 3x^2+ 2x+ 4\) which is \(\displaystyle 3\lim_{x\to a}x^2+ 2\lim_{x\to a} x+ \lim_{x\to a} 4= 3a^2+ 2a+ 4\).

Note: one thing you will soon learn, if you haven't already, is that, if f(x) is "continuous" at x= a is f(a). Often students come to the conclusion that "\(\displaystyle \lim_{x\to a} f(x)\)" is just a "fancy" way to say "f(a)". However, all of the really interesting limits are for f not continuous so that is not true!

 

[Steven G]

Yes, that is the most trivial limit problem there is! The problem "exits" pretty much to see if you were paying attention! It is also useful for more complicated problems such as "limit as x goes to a of \(\displaystyle 3x^2+ 2x+ 4\) which is \(\displaystyle 3\lim_{x\to a}x^2+ 2\lim_{x\to a} x+ \lim_{x\to a} 4= 3a^2+ 2a+ 4\).

Note: one thing you will soon learn, if you haven't already, is that, if f(x) is "continuous" at x= a is f(a). Often students come to the conclusion that "\(\displaystyle \lim_{x\to a} f(x)\)" is just a "fancy" way to say "f(a)". However, all of the really interesting limits are for f not continuous so that is not true!

To OP,
I suspect that Prof. Halls meant to say if f(x) is "continuous" at x= a, then \(\displaystyle \lim_{x\to a} f(x)\) is f(a)

 

[HallsofIvy]

Yes, thank you.

 

[Steven G]

lim x→3 4 = 4
Why there is a need for this example to exist if its numeric value is 4
I am not neglecting this situation, but why does it exist and where did it most likely come from?

Although when I first saw this type of limit it did not bother me, there were other obvious theorems that I could not believe that mathematicians would bother to prove. The one that stands out in my mind was the need to prove the intermediate value theorem, which you will learn about later this semester. Basically it says that if you drive you car from 50 mph to 60 mph then you had to drive at 58 mph at some point. If you asked anyone on the street corner they would say of course you had to go 58 mph at some point. Even though I understood the rigorous proof of the intermediate value theorem I thought it was a waste of time. That was back then! Now I have a much different attitude towards proven the obvious. First, I never assume anything is obvious anymore as I got burned too many times thinking that way (especially in probability theory!). 2ndly (especially if you are going to study advance math) I have learned that for completeness you must prove even the obvious theorems. I now feel that is very important, actually extremely important. My attitude is that I can not (and will not!) use a theorem if I can't prove it or at least understand the proof. I feel like that is cheating!
Just because, as Prof Halls stated, your limit is the most trivial limit does not mean that you should not be exposed to it. Beside you would not believe how many students do not know/understand this limit. They have trouble evaluating a constant function, say f(x) = 4 at x=11 (or in your case at x=3.99999) as it is quite abstract for some.

One last thing I feel I must say is that many students accept way too much from their math teachers without seeing why these facts are true. One can not just memorize mathematics, one must understand it! I have told my students at times to never believe anything that I say. I tell them to go home and verify to their satisfaction that what I said is in fact correct.