You've probably heard of vertical asymptotes, but do you know why they exist or what they
actually are?
To give you some idea, vertical asymptotes are kind of like fences built along a line
running through a certain point on the x-axis on the graph of a function.
The graph of a function can get close to the asymptote from either side, but it'll never
be able to cross or even touch it at all.
Perhaps we could think of it as an electric fence, or, since vertical asymptotes are usually
drawn as dotted lines, they're like invisible electric fences.
Let's call this line, the vertical asymptote, x=a, for some real number a.
Along the x-axis here, you can see that the graph of the function moves infinitely closer
to the vertical asymptote, but the actual value of x will never ever ever be equal to
a.
Even if we zoom in thousands of times, you'll find that the the two lines will never actually
touch.
Because they never touch, you may be imagining that the two lines are somehow parallel, but
in order to be parallel, these lines would need to have exactly the same slope.
The slope of the asymptote, though, is completely vertical, and the slope of the function is
constantly changing.
So the lines are definitely not parallel, but instead just infinitely close together.
Another question you may have when dealing with asymptotes is how they are different
from holes, or removable discontinuities.
Both are places on the function where the graph is discontinuous, right?
Well, let's take a look.
Here's a graph with an asymptote at x=3, and here's one with a hole at x=3.
Notice that for the hole, the function approaches a certain y-value, 2, as it approaches the
undefined x-value from both sides.
That is, the limit from both sides of the hole is the same.
In the function with the vertical asymptote, on the other hand, the function goes to positive
infinity as it approaches x=3.
So, the limit doesn't exist as x approaches the vertical asymptote, which makes vertical
asymptotes very different from removable discontinuities.
So, what we know now is that a vertical asymptote is not only a place on the graph where the
function is undefined, but also a place where the limit, as x approaches the vertical asymptote,
doesn't exist.
These facts also tell us that vertical asymptotes are non-differentiable, since differentiability
would require both continuity and the existence of a limit.
In the same way, vertical asymptotes also cannot be critical points or inflection points,
since one of the basic parameters for these points is to exist on the function.
Neither critical points nor inflection points ever occur at a discontinuity.
Now that we know what vertical asymptotes are and are not, the natural next question
is, when do these vertical asymptotes appear?
That is, how can we tell that a function will or will not have a vertical asymptote?
Most often, we run into vertical asymptotes when dealing with rational functions.
Rational functions, remember, are the ones that look like this, with a numerator and
denominator.
These functions have vertical asymptotes whenever they are undefined, and they are undefined
whenever their denominator is equal to zero, because of course dividing by zero can't be
done.
So if we're given a rational function, how do we actually find any vertical asymptotes
that it might have?
We can find any vertical asymptotes by setting the denominator equal to 0 and solving for
the variable.
In this case, we'd find that x=4, so there's a vertical asymptote along the line x=4 on
the graph.
Of course, not all rational functions have vertical asymptotes.
There are plenty of rational functions with denominators that are never equal to zero,
such as this equation, 1/(x^2+4).
If we set this denominator equal to zero and try to solve for x, we get that x is equal
to the square root of negative four, which is not a real number, and therefore, this
is not a vertical asymptote.
There are tons of examples of rational functions like this, so we can't ever assume that a
function has a vertical asymptote just because it's a rational function.
We also can't assume that just because a function isn't rational it doesn't have a vertical
asymptote.
In fact, there's another type of function that always has a vertical asymptote, and
that is the logarithmic function.
Looking at the graph of y=ln(x), it does look like as x approaches zero from the right side,
y approaches negative infinity.
Since logarithms are not defined for numbers that are not positive , there will be a vertical
asymptote at zero.
Other logarithmic functions, similarly, have vertical asymptotes wherever the input to
the function, or the argument, is not positive.
So for y=log_10(x+6), the vertical asymptote will be at -6, because at this point, the
input to the function would be log_10(0), which is undefined.
On the graph, the curve would never cross the line at x=-6.
So if a vertical asymptote is an invisible electric fence, we now know how to discover
that fence and understand why it exists.
To review, vertical asymptotes are places on the graph of a function where the function
is not defined and where the limit, as x approaches the asymptote, does not exist.
We also know that vertical asymptotes exist in both rational and logarithmic functions,
and we know how to find them in both.
In rational functions, we set the denominator of the function equal to zero and solve for
the variable.
And in logarithmic functions, we see where the argument of the function would be equal
to zero, and know that the vertical asymptote would be along that line.
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