Use Theorem 3.10 to evaluate the following limits.
lim x...

Question

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 7x) / (sin x)

Accepted Answer

Welcome back, everyone. Use the following theorem to evaluate the limit as X approaches 0 of tangent of 21 X divided by sin of X. It says limit SX approaches 0 of sin x divided by X is equal to 1. We're given for answer to A says -21, B says 0, C says 21, and D says 42. So, let's begin with the given limit. We're given limit as X approaches 0. In the numerator, we have tangent of 21 X and in the denominator, there's sign of X. If we attempt the right substitution, well, this is going to give us a 0 divided by 0, which is an indeterminate form. And essentially, we want to use the given limit to somehow somehow incorporate it and solve for the given limit. First of all, what we're going to do is rewrite the tangent function in terms of sine and cosine. So we're going to get sine of 21 X divided by cosine of 21 X, right, because tangent is the ratio between sine and cosine, and then we're going to rewrite sin of x in the denominator. So now from here we're going to distribute the limit. Essentially, we can notice that we can apply the limit for the top function and the bottom one. So we end up with limit. Of sin of 21 X as X approaches 0. Now we have a double division, right, so we can essentially write the product of two limits in the denominator. We're going to have limit as x approaches 0 of cosine. Of 21 X, which is multiplied by the limit as x approaches 0 of sin of x. Now what we can do from here is simply understand that. The limit as x approaches 0 of cosine of 21 X is simply cosine of 0, which is 1. So we can immediately get rid of that specific limit because it has no difference. It is equal to 1, and we can rewrite the ratio as limit. Now we can use a single limit. As x approaches 0 of sin of 21 X. Divided by sign of X. Now from here we're going to apply several manipulations. So what we're going to do is simply divide sign of 21 X by 21 X, right? So essentially what we're going to do is take sine 21 X, divide by 21 X, and this means that we also have to multiply it by 21 X to make sure that we're not changing the original function. So this is what we're going to do. And now for the bottom part we have sin x, so we're dividing it by X and multiplying it by X. The reason why we're choosing these exact multiples is simply because we want to have a form of sign of an angle divided by the same angle. So we're looking at each angle, and this is how we are choosing our multiples. So now notice that if we distribute the limit once again, we're going to get limit as X approaches 0 of 21 X. Multiplied by a sign of 21 X divided by 21 X. Right, so we have this one. And now notice that. Sign of 21 X divided by 21 X. If we apply a substitution such that 21 X is some angle alpha, well, this part is going to give us one, right? And then what we can do is still keep a single limit. The reason why we're going to do this and write a single limit is because we also want to cancel out those Xs. So we are going to write the fraction and at the bottom we have X, sine x divided by X. Now notice that we can cancel out the axes for each numerator, so we can cancel out X in front of sine X and X in front of sign of 21 X, right? And then let's label that each sign ratio with its corresponding angle is 1, right? So we have our first one, which is 1, and then our second limit which is 1. From here we can essentially factor out 21, right, because we have 21 remaining, then we end up with 21 limit as X approaches 0 of. Sign of 21 X divided by 21 X. This is going to have a limit of 1, and now we can distribute the limit for each part just to clearly see that we can apply the given definition. So the denominator becomes limit as x approaches 0 of sin of x divided by X. So both our numerator and the denominator gives us 1 for each limit, right? Because we have a sign of an angle divided by the same angle as X approaches 0. If X approaches 0, then 21 X will also approach 0. So overall, we can say that this ratio is 21 divided by 1, right? We can write 1 at the bottom as well. And that's it. We have evaluated the final answer, which corresponds to the answer choice C 21. Thank you for watching.

Use Theorem 3.10 to evaluate the following limits.
lim x🠂0 (tan 7x) / (sin x)

Key Concepts

Limits

Theorem 3.10 (L'Hôpital's Rule)

Trigonometric Limits

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