As x approaches infinity we only need to look at the highest order of polynomial in both the numerator and denominator. Then we compare the highest order polynomial in both the numerator and denominator. If the denominator is higher order our limit goes to zero, if the numerator is higher our order our limit goes to positive or negative infinity (depending on the sign of the highest order x term). If our numerator and denominator have the same order the limit goes to a/b where a is the coefficient for the highest order x in the numerator and b is the coefficient for the highest order x in the denominator. 

Our numerator has higher order and the coefficient for the x to the fourth term is negative so our limit goes to negative infinity. 

As x approaches infinity we only need to look at the highest order of polynomial in both the numerator and denominator. Then we compare the highest order polynomial in both the numerator and denominator. If the denominator is higher order our limit goes to zero, if the numerator is higher our order our limit goes to positive or negative infinity (depending on the sign of the highest order x term). If our numerator and denominator have the same order the limit goes to a/b where a is the coefficient for the highest order x in the numerator and b is the coefficient for the highest order x in the denominator. 

For our equation the orders are the same in x for the numerator and denominator (both 4). So we divide the coefficients of the highest order x terms to get our limit and we get

The function given is a polynomial with a term , such that  is greater than 1. 

Whenever this is the case, we can say that the whole function diverges (approaches infinity) in the limit as  approaches infinity.

This tells us that the given function is not a very realistic description of a car's speed for large !

First factor the numerator to simplify the function.

,

so

.

Now

.

There is no denominator now, and hence no discontinuity.  The limit can be found by simply plugging in  for .

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There are a number of ways to solve this. Either we can solve by finding what K(C) evaluates to for larger values of C and see where they converge, i.e. :

K(10000) = 952.38

K(1000000) = 999.5

etc...

And we see that K(C) approaches 1000 for larger concentrations, C.

Or we can notice that the dominant term in the numerator is 1000C; dominant term in the denominator is C.

1000C / C = 1000, which will ultimately be our limit.

To find    here, you need only plug in  for :

Find a common denominator for both the upper and lower expressions and then simplify:

Substituting the value of  will yield , which is not in indeterminate form.  Therefore, L'Hopital's rule cannot be used.

The question asks for the limit as x approaches to five from the right side in the graph. As the graph approaches closer and closer to , the y-value decreases to negative infinity and will never touch .

The correct answer is:  

In the unsimplified form, the limit does not exist; however, the numerator can be factored and simplified.

The first and only step is to substitute the value of  into the function.  Since there is not a zero denominator or an indeterminate form, we do not have to worry about L'Hopital or an undefined limit.

The limit will approach to .