Video Transcript
Compare the growth rate of the two
functions 𝑓 of 𝑥 equals 𝑥 plus four squared and 𝑔 of 𝑥 equals the natural log
of 𝑥 using limits as 𝑥 approaches ∞.
We begin by recalling that we can
use limits to compare the growth rate of two functions by using the following
definition. We ensure that 𝑓 of 𝑥 and 𝑔 of
𝑥 are positive for 𝑥 sufficiently large. Then if 𝑓 of 𝑥 grows faster than
𝑔 of 𝑥, the limit as 𝑥 approaches positive ∞ of the quotient 𝑓 of 𝑥 over 𝑔 of
𝑥 must itself be positive ∞. Or, equivalently, the limit as 𝑥
approaches positive ∞ of 𝑔 of 𝑥 over 𝑓 of 𝑥 would be equal to zero. Both 𝑓 of 𝑥 and 𝑔 of 𝑥 are
positive for sufficiently large values of 𝑥. In fact, the natural log of 𝑥 is
always positive and 𝑓 of 𝑥 is greater than or equal to zero for all values of
𝑥.
So we use the first part of our
definition. We’ll find the limit as 𝑥
approaches positive ∞ of 𝑥 plus four all squared over the natural log of 𝑥. Now, we notice that if we try
direct substitution, we obtain ∞ over ∞, which is, of course, of indeterminate
form. Instead then, we’re going to use
L’Hôpital’s rule. Now, this says that if 𝑎 is either
a finite number or ∞, if the limit as 𝑥 approaches 𝑎 of the quotient of 𝑓 of 𝑥
and 𝑔 of 𝑥 is either equal to zero over zero or ∞ over ∞, then the limit as 𝑥
approaches 𝑎 of 𝑓 of 𝑥 over 𝑔 of 𝑥 is equal to the limit as 𝑥 approaches 𝑎 of
𝑓 prime of 𝑥 over 𝑔 prime of 𝑥.
So let’s begin by finding the
derivative of 𝑓 and 𝑔. That’s 𝑓 prime of 𝑥 and 𝑔 prime
of 𝑥. Now, the general power rule says
that we can find the derivative of 𝑥 plus four all squared by multiplying 𝑥 plus
four by two. And then reducing the exponent by
one. And then multiplying all of that by
the derivative of the inner function. Well, the derivative of 𝑥 plus
four is one. So we have two times 𝑥 plus four
to the power of one times one, which is simply two times 𝑥 plus four. the
derivative of 𝑔, the derivative of the natural log of 𝑥, is a bit more
straightforward. It’s simply one over 𝑥. And so, the limit as 𝑥 approaches
positive ∞ of 𝑥 plus four all squared over the natural log of 𝑥 must be equal to
the limit as 𝑥 approaches positive ∞ of two times 𝑥 plus four over one over
𝑥.
Now, of course, dividing by a
fraction is the same as multiplying by the reciprocal of that fraction. So we get two times 𝑥 plus four
times 𝑥 over one, which is just two 𝑥 times 𝑥 plus four. And so, we see that as 𝑥
approaches ∞, two 𝑥 times 𝑥 plus four also approaches ∞. And referring back to our original
definition, we see that this means that 𝑓 of 𝑥 grows faster than 𝑔 of 𝑥. And so, the growth rate of 𝑓 of 𝑥
must be greater than the growth rate of 𝑔 of 𝑥.
