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Let's solve this problem step by step.

### Given:
We need to analyze the limit of the ratio of two polynomials \( P(x) \) and \( Q(x) \), as \( x \to \infty \):

\[
\lim_{x \to \infty} \frac{P(x)}{Q(x)}
\]

Where \( P(x) \) and \( Q(x) \) are polynomials with positive coefficients.

### Part (a): If the degree of \( P(x) \) is less than the degree of \( Q(x) \)

Let the degree of \( P(x) \) be \( n \) and the degree of \( Q(x) \) be \( m \), where \( n < m \). As \( x \to \infty \), the polynomial with the higher degree dominates the behavior of the ratio. Since \( Q(x) \) has a higher degree, \( Q(x) \) grows much faster than \( P(x) \).

Thus, the limit is:

\[
\lim_{x \to \infty} \frac{P(x)}{Q(x)} = 0
\]

### Part (b): If the degree of \( P(x) \) is greater than the degree of \( Q(x) \)

Now, let the degree of \( P(x) \) be \( n \) and the degree of \( Q(x) \) be \( m \), where \( n > m \). As \( x \to \infty \), the higher degree polynomial \( P(x) \) will dominate, meaning \( P(x) \) grows faster than \( Q(x) \).

In this case, the limit approaches infinity:

\[
\lim_{x \to \infty} \frac{P(x)}{Q(x)} = \infty
\]

### Summary of Results:

- **(a)** If the degree of \( P(x) \) is less than the degree of \( Q(x) \), the limit is \( 0 \).
- **(b)** If the degree of \( P(x) \) is greater than the degree of \( Q(x) \), the limit is \( \infty \).

Would you like any further details or have any specific questions? 

### 5 Related Questions:
1. What happens if the degrees of \( P(x) \) and \( Q(x) \) are equal?
2. How do these limits change if the coefficients of the polynomials are negative?
3. Can you explain how L'Hopital's Rule could be applied to this problem?
4. What happens to the limit if \( P(x) \) and \( Q(x) \) are not polynomials but rational functions?
5. How does the behavior of polynomials at infinity relate to their leading terms?

### Tip:
The behavior of polynomials at infinity is dictated entirely by their highest-degree term.

Let P and Q be polynomials with positive coefficients. Consider the limit below. Find the limit if the degree of P is less than the degree of Q, and find the limit if the degree of P is greater than the degree of Q.

Learn how to find the limit of the ratio of two polynomials with positive coefficients as x approaches infinity. This problem covers key calculus concepts such as limits, focusing on cases where the degrees of P and Q differ. Suitable for students in undergraduate calculus.

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