Limit's problems?

Question

1. what is the limit of x approaches infinity of (2ln(x)/x) ??

Answer 1

So we want to calculate

lim (2 ln(x)/x)
x -> infinity

First, let's pull the constant out of the limit. The wonderful thing about derivatives, integrals, and limits is that we can always take the constant out.

2 * lim (ln(x)/x)
x -> infinity

Note that this limit is of the form [infinity/infinity], so we can apply L'Hospital's rule. L'Hospital's rule states that, if a limit has the form [0/0] or [infinity/infinity], we can take the derivative of each of the numerator and denominator and still get the same limit.

The derivative of ln(x) is 1/x, and the derivative of x is 1. Thus, we get

2 * lim ((1/x)/1)
x -> infinity

Which reduces to become

2 * lim (1/x)
x -> infinity

For limits, the form [1/infinity] yields 0. So our answer is

2(0) = 0

Answer 2

lim of x approaches infinity of (2ln(x)/x) is 0/0 which puts us in an indeterminate form so we're allowed to use l'hospital's rule
take the limit of the the top and bottom differentiated seperately:
lim of x approaches infinity of 2/x which is 0

Answer 3

What happens as you replace x with larger and larger numbers?

x = 22,026.466... then f(x) = 0.000908
x = 59,874.142... then f(x) = 0.000367
and so on