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If $y=p(x)$ is a third degree polynomial function with real coefficients, then which of the following statements is not true for any such function?

  1. The range of $p$ is the set of all real numbers
  2. The graph of $p$ touches the x-axis in at least three different places.
  3. The graph of $p$ does not have any vertical or horizontal asymptotes.
  4. The graph of $p$ touches the x-axis in less than four different places

I know 2 is not true, but do not understand why 3 is correct, particularly why the graph of $p$ does not have any vertical asymptotes. Can anyone explain to me why choice 3 is correct.

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  • $\begingroup$ Consider what happens as x tends to plus or minus infinity $\endgroup$ – mrnovice Feb 26 '17 at 5:25
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let $p(x) = ax^{3} + bx^{2} + cx + d$

as $x$ tends to $\infty$, $p(x)$ tends to $\infty$ for $a >0$ and $-\infty$ for $a<0$

as $x$ tends to $-\infty$, $p(x)$ tends to $-\infty$ for $a >0$ and $\infty$ for $a<0$

Noting that $p(x)$ is continuous $\forall x$, there can't be any vertical or horizontal asymptotes. Note this fact is true for all polynomials of degree $2n+1$, $n \geq 1$

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Vertical asymptote means that $\lim_{x\to c}p(x)=\pm\infty$ for some finite $c$. This would mean that either $p$ is not defined at $c$, or $p$ is not continuous, both of which are untrue for polynomials in general.

An horizontal asymptote means that $\lim_{x\to \pm\infty}=c$ for some finite $c$. Again, it's easy to see that this is not true for polynomials in general (well, except for those of degree $0$ I guess).

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