1. Compute each of the following limits. \( \begin{array}{llll}\text { a) } \lim _{x \rightarrow \infty} 3 x^{4} & \text { c) } \lim _{x \rightarrow \infty}\left(-2 x^{5}\right) & \text { e) } \lim _{x \rightarrow \infty}\left(-\frac{2}{3} x^{6}\right) & \text { g) } \lim _{x \rightarrow \infty} 4 x^{3} \\ \text { b) } \lim _{x \rightarrow-\infty} 3 x^{4} & \text { d) } \lim _{x \rightarrow-\infty}\left(-2 x^{5}\right) & \text { f) } \lim _{x \rightarrow-\infty}\left(-\frac{2}{3} x^{6}\right) & \text { h) } \lim _{x \rightarrow-\infty} 4 x^{3}\end{array} \)

To compute each of the given limits, we can analyze the behavior of the polynomial functions as \( x \) approaches \( \infty \) or \( -\infty \). ### a) \( \lim _{x \rightarrow \infty} 3 x^{4} \) As \( x \) approaches infinity, \( 3x^{4} \) also approaches infinity. Therefore, the limit is: \[ \lim_{x \to \infty} 3 x^{4} = \infty \] ### b) \( \lim _{x \rightarrow -\infty} 3 x^{4} \) Since \( 3x^{4} \) is a polynomial of even degree, it approaches infinity as \( x \) approaches \( -\infty \): \[ \lim_{x \to -\infty} 3 x^{4} = \infty \] ### c) \( \lim _{x \rightarrow \infty} (-2 x^{5}) \) As \( x \) approaches infinity, \( -2x^{5} \) goes to negative infinity since the leading term is negative: \[ \lim_{x \to \infty} (-2 x^{5}) = -\infty \] ### d) \( \lim _{x \rightarrow -\infty} (-2 x^{5}) \) For \( x \) going to negative infinity, \( -2x^{5} \) becomes positive infinity because the negative lead of the term becomes positive: \[ \lim_{x \to -\infty} (-2 x^{5}) = \infty \] ### e) \( \lim _{x \rightarrow \infty} (-\frac{2}{3} x^{6}) \) As \( x \) goes to infinity, this limit approaches negative infinity due to the negative leading coefficient: \[ \lim_{x \to \infty} (-\frac{2}{3} x^{6}) = -\infty \] ### f) \( \lim _{x \rightarrow -\infty} (-\frac{2}{3} x^{6}) \) As \( x \) approaches negative infinity, \( -\frac{2}{3} x^{6} \) goes to positive infinity: \[ \lim_{x \to -\infty} (-\frac{2}{3} x^{6}) = \infty \] ### g) \( \lim _{x \rightarrow \infty} 4 x^{3} \) As \( x \) goes to infinity, this polynomial approaches infinity: \[ \lim_{x \to \infty} 4 x^{3} = \infty \] ### h) \( \lim _{x \rightarrow -\infty} 4 x^{3} \) For \( x \) approaching negative infinity, \( 4x^{3} \) approaches negative infinity: \[ \lim_{x \to -\infty} 4 x^{3} = -\infty \] In summary: \[ \begin{align*} \text{a)} & \quad \infty \\ \text{b)} & \quad \infty \\ \text{c)} & \quad -\infty \\ \text{d)} & \quad \infty \\ \text{e)} & \quad -\infty \\ \text{f)} & \quad \infty \\ \text{g)} & \quad \infty \\ \text{h)} & \quad -\infty \\ \end{align*} \]