Given $P(x)=5 x^{5}-2 x^{3}-4 x^{2}+8 x+4$,
$P(x) \rightarrow \square$ if $x \rightarrow-\infty$,
$P(x) \rightarrow \square$ if $x \rightarrow \infty$.
If your answer is $-\infty$, input -infinity; if your answer is $\infty$, input infinity.

Step 1 ：Given the polynomial function $P(x)=5 x^{5}-2 x^{3}-4 x^{2}+8 x+4$

Step 2 ：The limit of a polynomial function as x approaches infinity or negative infinity is determined by the term with the highest degree. In this case, the term with the highest degree is $5x^5$

Step 3 ：As x approaches negative infinity, $x^5$ will also approach negative infinity, and multiplying by 5 doesn't change the sign, so the limit as x approaches negative infinity is negative infinity

Step 4 ：As x approaches positive infinity, $x^5$ will also approach positive infinity, and multiplying by 5 doesn't change the sign, so the limit as x approaches positive infinity is positive infinity

Step 5 ：Final Answer: \(\boxed{\text{As } x \rightarrow -\infty, P(x) \rightarrow -\infty}\)

Step 6 ：Final Answer: \(\boxed{\text{As } x \rightarrow \infty, P(x) \rightarrow \infty}\)