Question

For rational limits:

when a big number such as ±[infinity] is approached in the denominator ______

Answer 1

For rational limits, as the denominator approaches infinity, the value of the fraction approaches zero. This behavior is associated with horizontal asymptotes in functions and is important for understanding limits involving infinity.

Step-by-step explanation:

For rational limits, when a big number such as ±[infinity] is approached in the denominator, the value of the fraction approaches zero. As the value in the denominator grows larger and larger, it causes the overall value of the rational expression to become smaller and smaller, eventually tending towards zero. This concept is vital for understanding different types of asymptotic behavior in functions, such as those seen in hyperbolic functions or when evaluating limits in calculus. In mathematics, when evaluating limits that involve infinity, this observation can act as a "guard rail" to help students check their work for potential errors or misapplications of mathematical operations like division or multiplication.

For instance, when looking at the function y = 1/x, as x approaches infinity, y approaches zero, which serves as a horizontal asymptote for the function. This principle also applies more broadly to fractions where the numerator has a fixed value, and the denominator heads towards infinity.