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Can you please help me with #25Please give all forms of the end behavior such as ups/downs, as_,_, and limits

Can you please help me with #25Please give all forms of the end behavior such as ups-example-1
User Mjaniec
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25) You have the following function:


f(x)=(x)/(2x+1)

Local behavior:

In order to determine the local behavior of f(x), let's find x and y-intercepts of the function. Furthermore, we can determine if there is a vertical asymptote in the graph.

The x-intercept is found for the value of x which makes the numerator of the function equal to zero. In this case, as you can notice, such a value is x = 0. Then, the x-intercept is the point (0,0).

The y-intercept is the value of the function where x=0:


f(0)=(0)/(2(0)+1)=0

Then, the y-intercept is the point (0,0).

A vertical asymptote is found for the value of x which makes the denominator of the function equal to zero. Then, in this case you have:


\begin{gathered} 2x+1=0 \\ 2x=-1 \\ x=-(1)/(2) \end{gathered}

Hence, there is a vertical asymptote for x =-1/2.

End behavior:

The end behavior is found by determining if there are horizontal or oblique asymtotes in the graph.

Take into account that if the degree of the numerator is the same that degree of the numerator, it means that there is a horizontal asymptote at y=a/b, where a and b are the leading coefficients of numerator and denominator respectvely.

In this case, both numerator and denominator have the same degree, then, there is a horizontal symptote for y = 1/2 (1 is the leading coefficient of numerator and 2 of denominator).

Now, take into account that oblique asymptotes occur when the degree of the denominator is one less than the degree of the numerator. This is not the case, then, there is no oblique asymptotes.

With the previous information about local and end behavior you can graph the given function as follow:

As you can notice, x and y-intercept are at point (0,0). Green line and purple lines represent vertical and horizontal asymptotes respectively.

Can you please help me with #25Please give all forms of the end behavior such as ups-example-1
User Piotr Tomasik
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