7. [ \( -/ 2 \) Points] DETAILS MY NOTES SCALCET9 2.6.055. Let \( P \) and \( Q \) be polynomials with positive leading coefficients. Consider the limit below. (If the limit is infinite, enter ' \( \omega \) ' or ' - \( \infty \) ', as appropriate. If the limit does not otherwise exist, enter DNE.) (a) Find the limit if the degree of \( P \) is less than the degree of \( Q \). (a) \( x(x) \) (b) Find the limit if the degree of \( P \) is greater than the degree of \( Q \). (b)

To solve the problem, we need to analyze the limits of the ratio of two polynomials \( P(x) \) and \( Q(x) \) as \( x \) approaches infinity. We will consider two cases based on the degrees of the polynomials. ### Case (a): Degree of \( P \) is less than the degree of \( Q \) Let: - \( \deg(P) = m \) - \( \deg(Q) = n \) - Given that \( m < n \) The limit can be expressed as: \[ \lim_{x \to \infty} \frac{P(x)}{Q(x)} \] Since \( P(x) \) has a lower degree than \( Q(x) \), as \( x \) approaches infinity, the leading term of \( Q(x) \) will dominate the behavior of the fraction. Therefore, we can conclude that: \[ \lim_{x \to \infty} \frac{P(x)}{Q(x)} = 0 \] ### Case (b): Degree of \( P \) is greater than the degree of \( Q \) Now, let’s consider the second case: - Given that \( m > n \) The limit can be expressed as: \[ \lim_{x \to \infty} \frac{P(x)}{Q(x)} \] In this case, since \( P(x) \) has a higher degree than \( Q(x) \), the leading term of \( P(x) \) will dominate the behavior of the fraction. Therefore, we can conclude that: \[ \lim_{x \to \infty} \frac{P(x)}{Q(x)} = \infty \] ### Summary of Results - (a) If the degree of \( P \) is less than the degree of \( Q \), the limit is \( 0 \). - (b) If the degree of \( P \) is greater than the degree of \( Q \), the limit is \( \infty \). Thus, the answers are: - (a) \( 0 \) - (b) \( \infty \)