Limit as x Goes to Infinity of x(1/x) | Answer Key - Edubirdie

University:
Massachusetts Institute of Technology
Course:
18.01SC | Single Variable Calculus
Academic year:

2022
Views:

60

Pages:

1
Author:

Shelby Chan

ex x and x x e Limits at Inﬁnity of ex x As x approaches inﬁnity, the rational expressions and x take on the form x e ∞ ∞ . Use the extended version of l’Hopital’s rule to evaluate the following limits, if they exist. ex x→+∞ x x b) lim x x→+∞ e a) lim Solution The extended version of l’Hopital’s rule tells us that: f (x) f � (x) = lim � x→+∞ g(x) x→+∞ g (x) lim provided that limx→+∞ f (x) and limx→+∞ g(x) have the appropriate properties � and limx→+∞ fg� ((xx)) exists. Since we have veriﬁed that the limits in question are of the form ∞ ∞ , we may apply this rule. a) ex x→+∞ x lim Because this is of the form ∞ ∞, we know that if the limit exists then: ex ex = lim = ∞. x→+∞ x x→+∞ 1 lim As x approaches inﬁnity, b) lim x→+∞ ex 1 does as well; hence so does ex x . x ex If the limit exists then: lim x→+∞ x 1 = lim x = 0. x→+∞ e ex As x approaches inﬁnity, the denominator of this rational function grows faster than the numerator and so the limit of the quotient is 0. We will soon see how limits such as these can be used to compare the rates of growth of two functions.

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