Mathematics

Problem

Answer the following questions related to the behaviour of exponential and polynomial functions for large positive values of

x

. a. (i) State the limits of

e^{x}

and

e^{-x}

x \to \infty

. b. (ii) Explain, without using any rule beyond the definition of limit, why

\displaystyle \lim_{x\to\infty} x^{3}e^{-x}=0

. c. (iii) Give one reason to justify that for every positive integer

n

\displaystyle \lim_{x\to\infty} x^{n}e^{-x}=0

. Your explanation should make clear the comparative growth of the two factors involved.