Video Transcript
Determine the limit of ln 𝑥 over
𝑥 as 𝑥 approaches infinity.
One way we might be tempted to
evaluate this limit is to use the fact that the limit of a quotient to functions is
the quotient of the limits of those functions. Applying this to our problem then,
we’d get the limit of ln 𝑥 as 𝑥 approaches infinity over the limit of 𝑥 as 𝑥
approaches infinity.
Now clearly, the limit in the
denominator is infinity. And it turns out that the limit in
the numerator is infinity. As the number gets bigger and
bigger without bound, so does the value of its logarithm. Unfortunately, infinity over
infinity is an indeterminate form, which gives us no insight into the value of the
limit that we want to evaluate. Both the limit in the numerator and
the limit in the denominator are infinite.
The way to evaluate this limit is
to use L’Hospital’s rule. This rule states that if either the
limit of 𝑓 of 𝑥 as 𝑥 approaches 𝑎 and the limit of 𝑔 of 𝑥 as 𝑥 approaches 𝑎
are both zero or if they are both infinite, then the limit of the quotient is the
limit of the quotient of their derivatives.
In our scenario, we want to
evaluate the limit of a quotient to functions. And we’ve seen that both the limits
of the function in the numerator is infinite. And the limit of the function in
the denominator is infinite. L’Hospital’s rule therefore
applies. And the limit of the quotient that
we’re looking for is the limit of the quotient of derivatives. We therefore need to evaluate the
limits of ln 𝑥 prime. That’s the derivative of ln 𝑥 with
respect to 𝑥 over 𝑥 prime. That’s the derivative of 𝑥 with
respect to 𝑥 as 𝑥 approaches infinity.
What is ln 𝑥 prime? The derivative of ln 𝑥 with
respect to 𝑥 is one over 𝑥. So let’s write that in. How about the derivative of 𝑥 with
respect to 𝑥. Well, that’s just one. One over 𝑥 divided by one is just
one over 𝑥. And so the limit we’re looking for
is just the limit of the reciprocal function one of 𝑥 as 𝑥 approaches
infinity. And this is a known limit to the
limit of the reciprocal function one over 𝑥 as 𝑥 approaches infinity is just
zero.
Before I finish the video, I should
just mention that there’re some small prints to L’Hospital’s rule. The functions 𝑓 and 𝑔 must both
be differentiable for the limit of the quotient of their derivatives to make
sense. And the derivative in the
denominator, 𝑔 power of 𝑥, must not be zero near the limit point 𝑎. But it can be zero at 𝑎. That’s okay. And finally, the limit on the
right-hand side of L’Hospital’s rule should exist or equal plus or minus
infinity. Or it’s not really much use.
In our case, the function 𝑓 was ln
𝑥. And this is of course
differentiable. Its derivative, we’ve seen, is one
over 𝑥. And 𝑔 was just the function 𝑥,
which of course is differentiable. Its derivative is one. And 𝑔 prime of 𝑥, which we saw
was one, is not equal to zero not even that near the limit point 𝑎, which was
infinity.
And finally, the limit of the
quotient of derivatives did exist because we found its value. It was zero.
