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 5x) / x

Accepted Answer

Welcome back, everyone. Use the following theorem to evaluate the limit as x approaches 0 of tangent of 14 X divided by X. We're given a limit as X approaches 0 of sin of x divided by X equals 1. And there are 4 answer choices A says -7, B 0, C 14, and D 28. Let's write down the limit that we want to evaluate that says limit as X approaches 0 in the num. we have agent of 14 X in the denominator we have X, and if we apply the right substitution, we're going to end up with an indeterminate form of 0 divided by 0. So we want to use the given theorem to evaluate the numerical value of this limit. And what we can do is simply a rewrite tangent in terms of sine and cosine, right? So let's recall that tangent is defined as the ratio between sine and cosine. So we get sine of 14 x divided by cosine of 14 X, and this represents the same tangent of 14 X. And now everything is divided by X. And what we can do is simply apply at right properties, and this is going to give us limit as x approaches 0, and we can actually group those terms, right? So we can decide what we want to divide. Well, we want to divide sine 14 X by X. Now what's remaining is the division by cosine of 14 X, so we can say that then we are dividing by cosine of 14X in our next step, or we can simply say that we're multiplying by the reciprocal of cosine of 14 X, and this gives us the same expression, right, because we have a double division so we can decide whether we want to divide it by cosine or by X first. And now from here, what we can do is simply distribute the limit for each. So we're going to get a limit. As x approaches 0 of sine of 14 of X divided by x multiplied by the limit. As x approaches 0 of 1 divided by cosine of 14 of X, and what we can notice is that if we substitute X equals 0 for cosine of 14 of X, well we're going to get 1 for that second limit. And this means that our goal is now to evaluate the limit as X approaches 0 of sine of 14 X divided by X. Now if we look at the theorem, we can interpret it as a sign of an angle divided by the same angle, right? So what we can do is simply replace that X with 14 of X, and this is going to give us the same angle of sign, right? But if we're dividing by 14, we also want to multiply by 14 to make sure that the original function is not changed. And now we're going to factor out the 14, and this gives us 14 limit as X approaches 0 of sine of 14 X divided by 14 X. And now if x approaches 0, apparently 14 X will also approach 0, so we're not changing the theorem, right? The theorem still holds, and we know that this particular limit is equal to 1 based on what's given. So the final answer is 14 multiplied by 1, which gives us 14, and the final answer to this problem corresponds to the answer choice C 14. Thank you for watching.

Use Theorem 3.10 to evaluate the following limits.
lim x🠂0 (tan 5x) / x

Key Concepts

Limits

Theorem 3.10 (Limit of a Trigonometric Function)

L'Hôpital's Rule

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