Find the limit as 𝑥 approaches 𝜋 of four times the sin of eight 𝑥.
We’re asked to evaluate the limit as 𝑥 approaches 𝜋 of a trigonometric function. And the first thing we should do when we’re asked to evaluate any limit is check if we can do this by using direct substitution. And in this case we know that we can evaluate the limit of any trigonometric function by using direct substitution. And it is important we do check this is only a trigonometric function. For example, we’re multiplying 𝑥 by eight and we’re multiplying the sin of eight 𝑥 by four. However, this is still just a trigonometric function. So we’re fine to use direct substitution.
Since our limit is as 𝑥 is approaching 𝜋, we just need to substitute 𝑥 is equal to 𝜋 into our function inside of our limit. Doing this, we get four times the sin of eight 𝜋. And one way of evaluating this is to recall that the sin function is periodic around two 𝜋. In other words, for any integer 𝑛 and any real value 𝑥, we know the sin of 𝑥 will be equal to the sin of 𝑥 plus two 𝑛𝜋. And we can see what happens when we set 𝑥 equal to zero and 𝑛 equal to four. We get that the sin of zero is the same as the sin of eight 𝜋.
And we know that the sin of zero is equal to zero. So we can evaluate the sin of eight 𝜋 to also give us zero. So we can evaluate this limit to just give us four times zero which is of course just equal to zero. And this is our final answer.
In this question, we were asked to evaluate the limit of a trigonometric function. And we know we can evaluate limits like this of trigonometric functions by using direct substitution. We just substituted 𝑥 is equal to 𝜋 into this trigonometric function. We then evaluated this and saw it was equal to zero.