For each of the following polynomial expressions, determine which monomial expression the polynomial expression tends to as $x \rightarrow \pm \infty$. a. $\left(5 x^{4}+x^{3}-12 x^{2}+x\right) \rightarrow$ Preview b. $\left(49-x^{2}\right) \rightarrow$ Preview c. $\left(-4 x^{2}+3 x^{6}-41 x+7\right) \rightarrow$ Preview

Step 1 ：The question is asking for the monomial expression that the polynomial expression tends to as x approaches positive or negative infinity. This means we need to find the term with the highest degree in the polynomial, because as x gets larger and larger (or smaller and smaller), the term with the highest degree will dominate the other terms.

Step 2 ：For example, in the polynomial \(5x^4 + x^3 - 12x^2 + x\), the term with the highest degree is \(5x^4\). As x approaches infinity, the value of \(5x^4\) will become much larger than the values of the other terms, so the polynomial will tend to \(5x^4\). The same logic applies when x approaches negative infinity.

Step 3 ：Final Answer: a. As \(x \rightarrow \pm \infty\), \(\left(5 x^{4}+x^{3}-12 x^{2}+x\right) \rightarrow \boxed{5x^{4}}\).

Step 4 ：b. As \(x \rightarrow \pm \infty\), \(\left(49-x^{2}\right) \rightarrow \boxed{-x^{2}}\).

Step 5 ：c. As \(x \rightarrow \pm \infty\), \(\left(-4 x^{2}+3 x^{6}-41 x+7\right) \rightarrow \boxed{3x^{6}}\).