As introduced in the earlier section, Introduction to Asymptotes:
DEFINITION
asymptote
An ‘asymptote’ (pronounced ASsimtote) is a
curve (usually a line)
that another curve gets arbitrarily close to as $\,x\,$ approaches $\,+\infty\,$, $\,\infty\,$, or a finite number. In particular, here is the definition of a vertical asymptote:
DEFINITION
vertical asymptote
A ‘vertical asymptote’ is a vertical line
that another curve gets arbitrarily close to
as $\,x\,$ approaches a finite number.
Specifically, the vertical line $\,x = c\,$ is a vertical asymptote for a function $\,f\,$ if and only if at least one of the following conditions is true:

The tangent function has infinitely many vertical asymptotes. For example, the vertical line $\,x = \frac{\pi}{2}\,$ is a vertical asymptote. as $\,x\rightarrow {\frac{\pi}{2}}^{+}\,$, $\,\tan x\rightarrow \infty\,$ as $\,x\rightarrow {\frac{\pi}{2}}^{}\,$, $\,\tan x\rightarrow \infty\,$ 
as $\,x\rightarrow 2^+\,$, $\,y \rightarrow \infty$  as $\,x\rightarrow 2^+\,$, $\,y \rightarrow \infty$  as $\,x\rightarrow 2^\,$, $\,y \rightarrow \infty$  as $\,x\rightarrow 2^\,$, $\,y \rightarrow \infty$  as $\,x\rightarrow 2^+\,$, $\,y \rightarrow \infty$ as $\,x\rightarrow 2^\,$, $\,y \rightarrow \infty$ 
as $\,x\,$ approaches $\,2\,$ from the right, $\,y\,$ goes to infinity 
as $\,x\,$ approaches $\,2\,$ from the right, $\,y\,$ goes to negative infinity 
as $\,x\,$ approaches $\,2\,$ from the left, $\,y\,$ goes to infinity 
as $\,x\,$ approaches $\,2\,$ from the left, $\,y\,$ goes to negative infinity 
How can a rational function (a ratio of polynomials) ‘blow up’ near a finite input?
When you're ‘trying’ to divide by zero!
Remember—dividing by a very small number gives a very big number.
Given a vertical asymptote, you usually want to know how the function behaves nearby.
Are the outputs going to infinity? To negative infinity?
Although you could certainly use a graphing calculator or WolframAlpha to see this behavior,
you should also be able to determine it algebraically, as shown next.
EXAMPLE:
The function $\displaystyle\,f(x) = \frac{3}{2x + 1}\,$ has a vertical asymptote
at $\,x = \frac 12\,$.
Why? The denominator, $\,2x + 1\,$, is zero when $\,x = \frac 12\,$, and the numerator is (always) nonzero.
Here is the thought process for determining what the outputs from $\,f\,$ look like,
close to $\,\frac 12\,$:
Consider values of $\,x\,$ a bit less than $\,\frac 12\,$ (i.e., just to the left of $\,\frac 12\,$):
Observe that the notation $\,\frac{()}{(\text{small })}\,$ is being used to denote a (normalsized) negative number, divided by a small negative number—which produces a large positive number. Next, consider values of $\,x\,$ a bit more than $\,\frac 12\,$ (i.e., just to the right of $\,\frac 12\,$):
Observe that we did not compute actual values of the output near $\,x = \frac 12\,$! There's no need—it would be working too hard. All we need to know is:

On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
