Graphs and Limits

Limits, one-sided limits, and infinite limits. Informal definitions. Calculus 1.

[akifcolak]

Firstly, thanks for your lessons, I want to mention about the infinity and limit notions. Many calculus books(Thomas, Stewart, Adams) state that if when we calculate limits, we see that the result is infinity, it does not mean that limit exists. Because infinity isnt a real number. Yes, maybe we can use infinity notation to just indicate the behavior but we have to say that limit doesnt exist(don't need to say a finite number). Actually, it makes sense because infinity word evokes the Limitless itself, such as there is no bound, it goes forever, not approach any number, any place etc.Besides, you also said L and a is real number in limit definition. Maybe you are aware of this explanation but I just want to say that.

[null-aleph]

After 2:22 minutes on this video, you said that "for any function f(x) and for real numbers a and L, lim_{x → a} f(x) = L means that f(x) gets arbitrarily close to L as x gets arbitrarily close to a. In other words, as x heads towards a, f(x) heads towards L." But are the statement "f(x) gets arbitrarily close to L as x gets arbitrarily close to a" and the statement "as x heads towards a, f(x) heads towards L" really the same? Didn't you mean "as close as we want to" there by the phrase "arbitrarily close"? Or you meant "closer and closer" by the phrase "arbitrarily close"?

When we talk about the definition of the limit of a function more informally, we can say the statement that "as x heads towards a, f(x) heads towards L" (which is similar to "as x gets closer and closer to a, f(x) gets closer and closer to L"). But when we talk about the definition of the limit of a function less informally, shouldn't we say that "f(x) gets arbitrarily close to L as x gets sufficiently close to a" (though you did mention it after a few seconds)? Are the statements "f(x) gets arbitrarily close to L as x gets arbitrarily close to a" and "f(x) gets arbitrarily close to L as x gets sufficiently close to a" same? Or f(x) gets arbitrarily close to L as x gets sufficiently close to a" is just a better way of saying the definition of the limit of a function than "f(x) gets arbitrarily close to L as x gets arbitrarily close to a"?

Is the statement "f(x) gets arbitrarily close to L as x gets arbitrarily close to a" appropriate for an informal definition of the limit of a function?

[xirbizos]

At 3:38 there is either a mistake or an inconsistency with my lecture:
In the definition I was taught, the limit only needs to consider what happens to "f(x)" when "x->a" within the interval(s) where "f" is actually defined.
So if "f" is not defined for values greater than "a", then the limit is still whatever "f(x)" approaches when "x->a" from the left side.