If , what value does approach as gets infinitely larger?
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The correct answer is E.
One way to determine the value that approaches as gets infinitely larger is to rewrite the definition of the function to use only negative powers of and then reason about the behavior of negative powers of as gets infinitely larger. Since the question is only concerned with what happens to as gets infinitely larger, one can assume that is positive. For , the expression is equivalent to the expression . As gets infinitely larger, the expression approaches the value , so as gets infinitely larger, the expression approaches the value . Thus, as gets infinitely larger, approaches .
Alternatively, one can use a graphing calculator to estimate the height of the horizontal asymptote for the function . Graph the function on an interval with “large” , say, from to .
By examining the graph, the all seem very close to . Graph the function again, from, say, to .
The vary even less from . In fact, to the scale of the coordinate plane shown, the graph of the function is nearly indistinguishable from the asymptotic line . This suggests that as gets infinitely larger, approaches , that is, .
Note: The algebraic method is preferable, as it provides a proof that guarantees that the value approaches is . Although the graphical method worked in this case, it does not provide a complete justification; for example, the graphical method does not ensure that the graph resembles a horizontal line for “very large” such as .