Video: Finding Limits Involving Trigonometric Functions

Find lim_(𝑥 → 0) 9𝑥/sin 6𝑥.

03:17

Video Transcript

Find the limit as 𝑥 tends to zero of nine 𝑥 divided by sin six 𝑥.

In order to solve this problem, we need to use two of the laws of limits. Firstly, if 𝑎 is a constant, then the limit as 𝑥 tends to zero of sin 𝑎𝑥 divided by 𝑥 is equal to 𝑎. Secondly, if 𝑓 of 𝑥 is the constant 𝑘 and 𝑏 exists in the real numbers, then the limit as 𝑥 tends to 𝑏 of 𝑓 of 𝑥 is equal to 𝑘.

We can rewrite our question the limit as 𝑥 tends to zero of nine 𝑥 divided by sin six 𝑥 as the limit as 𝑥 tends to zero of one divided by sin six 𝑥 divided by nine 𝑥. This, in turn, can be rewritten as the limit as 𝑥 tends to zero of one divided by sin six 𝑥 divided by 𝑥 multiplied by one-ninth.

We can then split this by taking the limit of the top, the numerator, and the bottom, the denominator, separately: the limit as 𝑥 tends to zero of one divided by the limit as 𝑥 tends to zero of sin six 𝑥 divided by 𝑥 multiplied by one-ninth.

Splitting the denominator using the laws of limits gives us the limit as 𝑥 tends to zero of one divided by the limit as 𝑥 tends to zero of sin six 𝑥 divided by 𝑥 multiplied by the limit as 𝑥 tends to zero of one-ninth. Using the second law that we wrote at the beginning of the question, if 𝑓 of 𝑥 is the constant 𝑘 and 𝑏 exists in the real numbers, then the limit as 𝑥 tends to 𝑏 of 𝑓 of 𝑥 is equal to 𝑘. This means the limit as 𝑥 tends to zero of one is equal to one. And likewise, the limit as 𝑥 tends to zero of one-ninth is equal to one-ninth.

Using the first law that we wrote down, if 𝑎 is a constant, then the limit as 𝑥 tends to zero of sin 𝑎𝑥 divided by 𝑥 is equal to 𝑎, we can see that the limit as 𝑥 tends to zero of sin six 𝑥 divided by 𝑥 is equal to six. This leaves us with one divided by six multiplied by one-ninth. Six multiplied by one-ninth is equal to six-ninths. So we have one divided by six-ninths. This is equal to nine-sixths or nine divided by six.

We can simplify this fraction to three-halves or three over two. Therefore, the limit as 𝑥 tends to zero of nine 𝑥 divided by sin six 𝑥 is equal to three over two or three-halves.

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