Final answer:
For a rational limit evaluated at infinity with the degree of the numerator equal to the degree of the denominator, the limit is the ratio of the leading coefficients.
Step-by-step explanation:
When evaluating the limit of a rational function as x approaches infinity, if the degree of the numerator equals the degree of the denominator, the limit of the function will be the ratio of the leading coefficients of the numerator and the denominator.
To find this limit, one usually divides every term in the numerator and the denominator by the highest power of x found in the fraction. As x approaches infinity, all terms with powers of x will tend to zero, leaving only the constants. Assuming the leading coefficient of the numerator is a and that of the denominator is b, the limit will therefore be a/b.
This property holds true because, in mathematics, we are allowed to perform the same operations on both sides of an equality, and since these terms in the numerator and denominator with powers of x tend towards zero, they effectively become negligible, resulting in a constant value as x approaches infinity.