In calculus, finding the limit of a function as \( x \) approaches infinity or negative infinity is a common task, particularly with rational functions. The key to solving these limits efficiently lies in identifying the highest power of \( x \) in the numerator and denominator.

When evaluating limits at infinity, the behavior of the function is determined by the degrees of the polynomials involved. If the highest power of \( x \) in the numerator and denominator are equal, the limit can be found by taking the ratio of the leading coefficients. For instance, if we have a function where both the numerator and denominator are polynomials of degree \( n \), the limit as \( x \) approaches infinity can be expressed as:

\[ \lim_{x \to \infty} \frac{a_n x^n + \ldots}{b_n x^n + \ldots} = \frac{a_n}{b_n} \]

where \( a_n \) and \( b_n \) are the leading coefficients of the numerator and denominator, respectively.

In the first example, we find the limit as \( x \) approaches negative infinity for the function \( \frac{7x^2}{49x^2} \). Here, both the numerator and denominator have the highest power of \( x^2 \). The limit simplifies to:

\[ \lim_{x \to -\infty} \frac{7}{49} = \frac{1}{7} \]

In the second example, we consider the function \( 5x^2 + 3 \). By rewriting it as \( \frac{5x^2 + 3}{1} \), we see that the highest power of \( x \) is in the numerator. As \( x \) approaches negative infinity, the function tends to infinity, indicating that the limit does not exist.

For the third example, with a rational function where both the numerator and denominator have the highest power of \( x^{10} \), the limit as \( x \) approaches positive infinity is determined by the coefficients of these terms. Thus, we find:

\[ \lim_{x \to \infty} \frac{10x^{10}}{23x^{10}} = \frac{10}{23} \]

In summary, when evaluating limits at infinity for rational functions, remember to identify the highest powers of \( x \) in both the numerator and denominator. If the degree of the numerator is greater, the limit approaches infinity; if the degree of the denominator is greater, the limit approaches zero. If the degrees are equal, the limit is the ratio of the leading coefficients. This method provides a quick and effective way to solve these types of problems.