### Video Transcript

Determine the limit as 𝑥 approaches zero of the sin of four 𝑥 over five divided by the tan of two 𝑥.

In this question, we’re asked to evaluate the limit of the quotient of trigonometric functions. And since we know we can evaluate the limits of trigonometric functions by using direct substitution, we can attempt to evaluate this limit by direct substitution. However, if we substitute 𝑥 is equal to zero into this function, we get the sin of four times zero over five divided by the tan of two times zero, which simplifies to give us the sin of zero divided by the tan of zero.

If we evaluate the numerator and denominator, we get zero divided by zero, which is an indeterminate form, which tells us we can′t evaluate this limit by using direct substitution. Instead, since we′re evaluating the limit of the quotient of two trigonometric functions, let′s start by recalling two of our useful trigonometric limit results.

We recall for any real constant 𝑎 the limit as 𝑥 approaches zero of the sin of 𝑎𝑥 divided by 𝑥 is equal to 𝑎. Similarly, for any real constant 𝑎, the limit as 𝑥 approaches zero of the tan of 𝑎𝑥 divided by 𝑥 is also equal to 𝑎. Currently, however, our limit is not in a form where we can directly use these results since both of these results involve a quotient with 𝑥. So to use these results, we′re going to need to introduce a quotient with 𝑥. We’ll do this by multiplying both the numerator and denominator of our expression through by 𝑥.

And it′s worth noting this won′t change the value of this limit since 𝑥 divided by 𝑥 is equal to one unless 𝑥 is equal to zero. However, we′re taking the limit as 𝑥 approaches zero, so we know our value of 𝑥 is never equal to zero. This then gives us the limit as 𝑥 approaches zero of 𝑥 times the sin of four 𝑥 over five all divided by 𝑥 multiplied by the tan of two 𝑥. Now to evaluate this limit by using our limit results, we′re going to want to split our limit. And to do this, we′re going to need to use the product’s rule for limits. We split this function into 𝑥 divided by the tan of two 𝑥 multiplied by the sin of four 𝑥 over five divided by 𝑥.

The product’s rule for limits tells us the limit of a product is equal to the product of the limits provided the limits of both factors exist. Therefore, we can rewrite our limit as the limit as 𝑥 approaches zero of 𝑥 divided by the tan of two 𝑥 multiplied by the limit as 𝑥 approaches zero of the sin of four 𝑥 over five divided by 𝑥. And now, we′re almost in a place where we can directly evaluate this limit. For example, we can already evaluate the second limit by using our limit result. Our value of 𝑎 is four over five. So, our limit result tells us that this limit evaluates to give us four-fifths.

We can′t yet directly evaluate our first limit since the tangent function is in the denominator. However, in our limit result, the tangent function is in the numerator, although if we take the reciprocal of both sides of the equation of our limit result, we can use the power rule for limits to write the tangent function in the denominator.

By the power rule for limits, we can take the reciprocal inside of our limit provided the limits exist. This gives us the limit as 𝑥 approaches zero of 𝑥 divided by the tan of 𝑎𝑥 is equal to one divided by 𝑎 provided 𝑎 is not equal to zero. And in our case, the value of 𝑎 is two. So, we can evaluate this limit to be one-half. And it′s worth noting since we′ve shown both of these limits exist, this justifies our use of the product’s rule for limits. And finally, we can evaluate this limit. One-half multiplied by four-fifths is two over five.

Therefore, we were able to show the limit as 𝑥 approaches zero of the sin of four 𝑥 over five divided by the tan of two 𝑥 is equal to two-fifths.