For infinity limits, we only consider the leading exponents (with their respective coefficients).  Therefore, our complicated functions simplify to the following:

When we plug in more and more negative values, this will keep getting bigger, and the negative signs will cancel each other out.  Therefore, this will tend to positive infinity.

For infinity limits, we need only consider the leading exponents.  Everything else will not matter when we plug in negative infinity. Therefore, our complex limit becomes much simpler:

Because this is an odd exponent and we are plugging in increasingly more negative values, the function will tend to larger and larger negative values. 

Horizontal asymptotes occur when the degree of the polynomial matches between the numerator and denominator or when the degree of the denominator polynomial is less than the numerator degree. The horizontal asymptote in this case is the  axis.

Since we are looking for a horizontal asymptote, this refers to a  value.

Look at a graph of .

At , there is a vertical asymptote.

The graph goes down to  as .

If a function is to be everywhere differentiable, then it must also be continuous everywhere. This implies that the one sided limits must be equal: .

Next, the one sided limits of the derivative must also be equal. That is, .

Now we have two missing variables and two equations. Set up a system and solve by elimination or substitution. 

 For a function to be continuous at a point ,  must exist and . 

This is true for all values of  except and .

Therefore, the interval of continuity is .

In order to show this statement is false, we must provide one counterexample. 

For example, let . Both of these are continuous functions (Their graphes are connected with no "jumps" or "breaks"). Then , but  does not have an absolute maximum, or mimimum.

The piecewise function

indicates that  is one when  is less than five, and is zero if the variable is greater than five.  At , there is a hole at the end of the split.  

The limit does not indicate whether we want to find the limit from the left or right, which means that it is necessary to check the limit from the left and right.  From the left to right, the limit approaches 1 as  approaches negative five.   From the right, the limit approaches zero as  approaches negative five.

Since the limits do not coincide, the limit does not exist for .

For a function to be continuous at a particular point, the limit of the function at that point must be equal to the value of the function at that point. 

 First, notice that 

. 

This means that the function is continuous everywhere.

Next, we must compute the limit. Factor and simplify f(x) to help with the calculation of the limit.

Thus, the limit as x approaches three exists and is equal to , so I and II are true statements. 

The definition of continuity at a point is

.

Since  is in the interval of the domain of the function, we see 

Also, since the function approaches  as  approaches  from the left and the right, we see that

.

Since 

the function is continuous at .