Use the graph of $h(x)=x^{3}-4 x$ to find the following limits. If necessary, choose the most informative answer from $\infty,-\infty$, and "Does Not Exist." (a) $\lim _{x \rightarrow \infty} h(x)=\llbracket$ (b) $\lim _{x \rightarrow-\infty} h(x)=$

Step 1 ：The problem is asking for the limit of the function \(h(x)=x^{3}-4 x\) as \(x\) approaches \(\infty\) and \(-\infty\).

Step 2 ：The limit of a function as \(x\) approaches \(\infty\) or \(-\infty\) is the value that the function approaches as \(x\) gets arbitrarily large or small.

Step 3 ：For the function \(h(x)=x^{3}-4 x\), as \(x\) gets larger and larger (approaches \(\infty\)), the \(x^{3}\) term will dominate the \(-4x\) term, because the power of \(x\) in \(x^{3}\) is larger than in \(-4x\). Therefore, as \(x\) approaches \(\infty\), \(h(x)\) will also approach \(\infty\).

Step 4 ：Similarly, as \(x\) gets smaller and smaller (approaches \(-\infty\)), the \(x^{3}\) term will still dominate the \(-4x\) term. However, because the power of \(x\) in \(x^{3}\) is odd, the sign of \(x^{3}\) will be negative when \(x\) is negative. Therefore, as \(x\) approaches \(-\infty\), \(h(x)\) will approach \(-\infty\).

Step 5 ：Final Answer: (a) \(\lim _{x \rightarrow \infty} h(x)=\boxed{\infty}\)

Step 6 ：Final Answer: (b) \(\lim _{x \rightarrow-\infty} h(x)=\boxed{-\infty}\)