Science:Math Exam Resources/Courses/MATH100/December 2015/Question 10 (c)/Solution 1

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When , The rate at which the exponential function decays to zero is much faster than the rate at which the two linear functions and increase, therefore

Indeed, more rigorously, it follows from L'Hospital's rule:


On the other hand, when , using the graph of the functions we can easily see that

and .

Therefore, has a horizontal asymtote only when which is at the very end of positive x-axis.


Horizontal Asymptote: