Answer :
The given function is f(x) = [tex]4x^6 + 19x^3[/tex].
The function does not have any horizontal asymptotes.
The function does not have any vertical asymptotes.
To find the horizontal asymptotes of the graph of the function, we need to analyze the behavior of the function as x approaches positive and negative infinity.
As x approaches positive infinity, the highest power of x dominates the function. In this case, the highest power of x is [tex]x^6[/tex]. Since [tex]x^6[/tex] approaches infinity as x goes to infinity, the function also approaches infinity.
Similarly, as x approaches negative infinity, the highest power of x dominates the function. Again, the highest power of x is [tex]x^6[/tex]. Since [tex]x^6[/tex] approaches infinity as x goes to negative infinity, the function also approaches infinity.
Therefore, the function does not have any horizontal asymptotes.
To find the vertical asymptotes of the graph of the function, we need to analyze the behavior of the function as x approaches the values that make the denominator equal to zero. However, in this case, there is no denominator in the given function. Therefore, the function does not have any vertical asymptotes either.
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