To find any horizontal asymptotes of the function y=(x³ +3x+5)/(100x²-18), we need to determine the limit of the function as 'x' approaches positive and negative infinity.
For a rational function like this one, where the degree of the numerator is greater than the degree of the denominator, the limit as 'x' approaches infinity is determined by the highest degrees of 'x' in the numerator and the denominator.
1. Firstly, let's find the limit as 'x' approaches positive infinity. The degree of 'x' in the numerator is three and in the denominator is two. Since the degree of the numerator is greater, as 'x' increases toward positive infinity, the function will tend to positive infinity. Therefore, the limit as 'x' approaches positive infinity is infinity, denoted as y=∞.
2. Now, let's find the limit as 'x' approaches negative infinity. Similarly, as the degree of 'x' in the denominator is less than that in the numerator, as 'x' decreases toward negative infinity, the function will tend towards negative infinity. Therefore, the limit as 'x' approaches negative infinity is negative infinity, denoted as y=-∞.
Hence, for the function y=(x³ +3x+5)/(100x²-18), the horizontal asymptotes are y=∞ for positive x-values and y=-∞ for negative x-values. These represent the values that the function gets arbitrarily close to but never actually reaches as 'x' gets infinitely large in either the positive or negative direction.