Question

Find the vertical asymptotes (if any) of the graph of the function. (Use \( n \) as an arbitrary integer if necessary, If an answer does not exist, enter DNE.) \[ h(x)=\frac{x^{2}-25}{x^{3}+5 x^{2}-x-

Answer 1

The vertical asymptotes of the graph of the function
`\( h(x) = (x^2 - 25)/(x^3 + 5x^2 - x - n) \)` are ( x = -5 ) and ( x = 1 ).

Step-by-step explanation:

To find the vertical asymptotes, we need to examine the denominator of the rational function since vertical asymptotes occur where the denominator equals zero. In this case, the denominator is
`\( x^3 + 5x^2 - x - n \)`. Setting this equal to zero and solving for ( x ), we can factor the denominator:

`\[ x^3 + 5x^2 - x - n = 0 \]`

`\[ (x + 5)(x - 1)(x + n) = 0 \]`

Setting each factor equal to zero gives the possible values for
`\( x \): \( x = -5, \, x = 1, \) and \( x = -n \)`. However, we are asked to express the answer without ( n ) if possible. Since ( n ) is an arbitrary integer, it can take any value, and ( x = -n ) will vary accordingly. Therefore, we exclude ( x = -n ) from the final answer to provide a concise solution.

Therefore, the vertical asymptotes of the function are ( x = -5 \k) and ( x = 1 ). These are the values of ( x ) where the denominator becomes zero, resulting in an undefined fraction and the presence of vertical asymptotes in the graph of the function.