There are three types of asymptotes; horizontal,vertical, and slant.

(1) Vertical asymptotes occur in rational functions when the rational function, written with no common factors in the numerator and denominator, has a factor in the denominator that is zero.

Factoring the expression we get:

`(x(x-7))/(x^3-49x)=(x(x-7))/(x(x+7)(x-7))`

Note the common factor of x-7 -- the function will not be defined at x=7, but there will not be a vertical asymptote there. Most graphing utilities will not even show the "hole" that occurs when x=7.

Also, there is a common factor of x in both numerator and denominator. Again, there will not be a vertical asymptote at x=0.

There will be a vertical asymptote at x=-7.

(2) Horizontal asymptotes occur in rational functions if:

(a) The degree of the numerator is less than the degree of the denominator. Then the horizontal asymptote is y=0.

(b) If the degree of the denominator is equal to the degree of the numerator then the horizontal asymptote occurs at `y=a/b` where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator.

(c) If the degree of the numerator is greater than the degree of the denominator there is no horizontal asymptote, but there may be a slant asymptote.

Here the degree of the numerator is smaller than the degree of the denominator so the horizontal asymptote is y=0.

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There is a vertical asymptote at x=-7. (The function is undefined when x=0 or x=7.)

There is a horizontal asymptote at y=0.

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The graph:

** Note that the function is not defined at x=0 or x=7, despite the apparent value in the graph. There are "holes" in the graph at those points.

The two types of asymptotes are vertical and horizontal.

Vertical asymptotes happen when there is an invalid `x` value (most likely causing the denominator to be zero).

Horizontal asymptotes happen when a certain `y` value cannot be reached.

First, let's simplify the function:

`(x(x-7))/(x^3-49)`

Factor out an x from the denominator and factor the difference of squares:

`(x(x-7))/(x(x+7)(x-7))`

To find vertical asymptotes, we need to find out what values would make the denominator zero.

By setting `x` , `(x+7)` , and `(x-7)` to zero, we find that these x values are `0, -7, 7`.

To find horizontal asymptotes, let's simplify the function again.

We can cancel out `x` and `(x-7)` , resulting in:

`1/(x+7)`

This is just a translation of the inverse function `1/x` , a function which has a horizontal asymptote of zero.

So the horizontal asymptote is `y=0`

Therefore, the vertical asymptotes of this function are:

`x=0`

`x=7`

` ` `x=-7`

And the horizontal asymptote is:

`y=0`