Video Transcript
Determine the following infinite limit: the limit as 𝑥 approaches zero from the right of five times the natural logarithm of the sin of 𝑥.
In this question, we’re given a limit to evaluate. And we’re told that this is an infinite limit. And before we recall what we mean by an infinite limit, we can start by noticing this is the limit as 𝑥 approaches zero from the right. This means our values of 𝑥 are going to approach zero. However, our values of 𝑥 will always be positive. They need to approach zero from the right.
We can now discuss what it means for a limit to be infinite as 𝑥 approaches zero from the right. One of two things must happen to our function in this case. Either the outputs will increase without bound or the outputs will decrease without bound. This is what it means to have an infinite limit. For example, in the first case, when the outputs increase without bound as 𝑥 approaches zero from the right, we say the limit is positive ∞, or in other words just ∞. And in the other case, when the outputs are decreasing without bound, we say that this limit is negative ∞.
And it’s worth reiterating here these are not the only ways a limit can be undefined. For example, we can have oscillating behavior, or we can have a combination: oscillating behavior where the size of the outputs are unbounded. However, these are the only two cases where we can say we have an infinite limit.
And there are many different ways of determining which of these cases we’re in for this limit. One way of doing this is to use a substitution. We’re going to use the substitution 𝑦 is equal to the sin of 𝑥 because then our limit will just be of the natural logarithm function. And we can then just use any results we know about the natural logarithm function. To substitute 𝑦 is equal to the sin of 𝑥 into this limit, we first need to determine what happens to 𝑦 as 𝑥 approaches zero from the right.
And there are a few different ways of doing this. We’re going to look at this graphically. We can sketch the curve of 𝑦 is equal to the sin of 𝑥 and determine what happens to the values of 𝑦 as 𝑥 approaches zero from the right. As 𝑥 approaches zero from the right, we can see that our output values of 𝑦 are also approaching zero. However, we can gain more information than this from the diagram. We can see our output values are always above the horizontal axis. In other words, the output values are always positive. As 𝑥 approaches zero from the right, 𝑦 also approaches zero from the right.
Therefore, we can now use the substitution 𝑦 is equal to the sin of 𝑥 to rewrite our limit. We get the limit as 𝑥 approaches zero from the right of five times the natural logarithm of the sin of 𝑥 is equal to the limit as 𝑦 approaches zero from the right of five times the natural logarithm of 𝑦. And now this is a much easier limit to evaluate. First, we’ll take the constant factor of five outside of our limit. This gives us five times the limit as 𝑦 approaches zero from the right of the natural logarithm of 𝑦.
And now there’s many different ways we can evaluate this limit. One way is to recall the following fact for the limits of the natural logarithm function. The limit as 𝑥 approaches zero from the right of the natural algorithm of 𝑥 is negative ∞. And we can determine this directly from the graph of the natural logarithm function. As our values of 𝑥 approach zero from the right, we can see the output values of 𝑦 are going to approach negative ∞. They’re decreasing without bound.
Therefore, the limit as 𝑥 approaches zero from the right of the natural logarithm of 𝑥 is negative ∞. We can use this to evaluate our limit. However, we do need to be slightly careful because we are multiplying this by a constant value of five. And in this case, it won’t change our answer because if something is decreasing without bound, multiplying it by five will also allow it to decrease without bound. It won’t change anything. However, if we multiplied it by a negative number, it would no longer be decreasing. It would in fact be increasing. So it is important to check the sign of this constant value.
Therefore, we were able to show the limit as 𝑥 approaches zero from the right of five times the natural logarithm of the sin of 𝑥 is negative ∞.
