Limits

Question

Evaluate the following limits: $$ \lim_{x \to \infty} e^{-x} $$ $$ \lim_{x \to \infty} xe^{-x} $$ $$ \lim_{x \to \infty} x!e^{-x} $$
Where $x!$ is the factorial of $x$, i.e. $x*(x-1)*(x-2)*(x-3)*...*3*2*1$

Hints

For limits that are the product of two terms, think about whether one term grows faster than the other term shrinks. As we are looking at the limit as $x$ approaches inifinity, consider what happens for large values of $x$.

Answer

$$ \lim_{x \to \infty} e^{-x} = 0 $$ $$ \lim_{x \to \infty} xe^{-x} = 0$$ $$ \lim_{x \to \infty} x!e^{-x} = \infty $$
For the last limit, consider large values of $x$ - for example, $x=100$. Then, compare to $x=101$. $$ 101!e^{-101} = 101 e^{-1} * 100!e^{-100} > 100!e^{-100} $$ This pattern follows for larger values of $x$, and so the function is strictly increasing, and will approach infinity as $x$ approaches infinity.