Evaluate the limit as $x\to\pm\infty$ and identify any horizontal asymptotes for the function $f\left(x\right)=\frac{x^4-25}{x^2-25}$.

$\displaystyle \lim_{x \to \infty}{f(x)}=\lim_{x \to -\infty}{f(x)}=\infty$

No horizontal asymptotes

$\displaystyle \lim_{x \to \infty}{f(x)}=\lim_{x \to -\infty}{f(x)}=-\infty$

No horizontal asymptotes

$\displaystyle \lim_{x \to \infty}{f(x)}=\lim_{x \to -\infty}{f(x)}=1$

Horizontal asymptote: $y=1$

$\displaystyle \lim_{x \to \infty}{f(x)}=1$; $\displaystyle \lim_{x \to -\infty}{f(x)}=-1$

Horizontal asymptotes: $y=1$ and $y=-1$