Video: Finding the Limit of Trigonometric Functions

Determine lim_(𝑥 → 8) (ln 𝑥)/𝑥.

03:25

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 two [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.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.