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

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When , the linear functions and become very large, however the exponential function decays very faster and becomes zero, 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: