Question 1

for limits at infinity where f(x) is unbounded- how do we determine (not visually) if the limit is infinity or negative infinity?

Accepted Answer

You can get a vertical asymptote when you have a rational function where your variable is in the denominator, and it is possible to set that variable such that the denominator evaluates to zero.  Think about it, as you pick values for the variable that get the denominator closer and closer to zero, the bigger the magnitude of the function -- the value of the function is going to go to positive or negative infinity as the denominator goes to zero.  This is a vertical asymptote.

Thus the function f(x) = 1/(x^2 + 4x + 3) has two vertical asymptotes, one at x = -1 and one at x = -3.  As these are the two values of x that cause the denominator to get to zero.  How do I know that?  The denominator can be factored:  f(x) = 1/((x + 1)(x + 3)

Now it is easy to see that setting x to -1 or -3 causes the denominator to go to zero.

There is one caveat.  If the denominator contains a factor that is also in the numerator, the x value that would cause that factor to be zero, and thus make the whole denominator be zero will NOT cause a vertical asymptote, it will cause a hole in the function.  Example:

Let f(x) = (x^2 + 4x + 3)/(x + 1).  I can factor this to:
f(x) = (x + 1)(x + 3)/((x + 1).  Algebraically I can cancel the (x + 1) factors from the numerator and denominator and end up with a linear function:  g(x) = (x + 3).  Because I can algebraically cancel the (x + 1) factor from the numerator and denominator, setting x to -1 to make that factor zero will NOT cause a vertical asymptote in the original function, it will cause a hole in the function.

Question 2

What does f(x) is unbounded mean?

Thanks

Accepted Answer

In a domain *D*, a function f is bounded when a real number M exists to make ```
 |f(x)|≤M 
``` 
If M is not defined (i.e. infinite), then f(x) is unbounded.

For example, sin x is bounded because |sin x |≤1, while 1 is its upper bound and -1 is its lower bound.
In this video, the function can be made bounded if the domain is, for example, (2, 8) or (−3, 1).

Question 3

let's say we defined x=2 to have f(x)= -4. would x=2 still be an asymptote? (considering that approaching x=2 from the left side will bring us to infinity)

Accepted Answer

Yes.  We can define a vertical asymptote of a function f(x) to occur at x = a if a one-sided limit of f(x) as x-->a is positive or negative infinity (if it behaves that way from both sides of a, that's okay too).  But as always, the limit doesn't care at all about what happens *at* x = a.  So if the function happens to be defined as `f(2) = -4` there but blows up infinitely on one side, it would still be considered a vertical asymptote at x = 2.  Hope that helps!

Question 4

What is an unbounded limit?

Accepted Answer

Unbounded limit is infinite limit. This is when x->a, f(x)-> ±∞. Such is an example of a vertical asymptote where f(x) -> ±∞. This is called unbounded limit or infinite limit. Though we call it infinite limit. This is a non-existing limit or the limit does not exist (dne). For the limit to exist, x->a, f(x)>#. Since ±∞ is not a #, so its limit dne. But if it dne, why do we say it is infinite limit? Though it dne, we know its end-behavior is approaching ±∞. So instead of saying it dne, it's more accurate to say its end-behavior.

Question 5

Hi! I liked the video; I found it mostly helpful for understanding the concepts. However, I do have one question: the video is saying that the limit of a graph as it approaches an asymptote is positive or negative infinity, but why isn't the asymptote the limit? Isn't a limit the phenomenon of which a graph is approaching a certain point?

Accepted Answer

The limit is the value that the function approaches, which in this case if f(x) or y. The asymptote is the line representing the value that the input of the function, or x, is approaching. As x approaches negative four, y approaches infinity. Since y is the function, infinity is the limit at that input.

Question 6

Can we find a function's asymptote algebraically?

Accepted Answer

A vertical asymptote occurs where the function is undefined (e.g., the function is y=A/B, set B=0). A horizontal asymptote (or oblique) is determined by the limit of the function as the independent variable approaches infinity and negative infinity.

Algebraically, there are also a couple rules for determining the horizontal (or oblique asymptote).
If the degree of the numerator is higher than the denominator, an oblique asymptote occurs. When the numerator and denominator have the same degree, the horizontal asymptote is the fraction of the leading coefficients of highest degree term. If the degree of the denominator is less than the degree of the numerator, the function will have a horizontal asymptote as y=0 (because the limit of the functions approaches 0).

Course: Calculus, all content (2017 edition) > Unit 1

Visually determining vertical asymptotes (old)

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