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

Question

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (sin 7x) / 3x

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Use the following theorem to evaluate the limit as X approaches 0 of sin of 39 x divided by 9 X. The limit as X approaches 0 of sin of x divided by X is equal to 1. Awesome. So it appears for this particular problem we're asked to use the following theorem, which the theorem is given to us as the limit as x approaches 0 of sin of x divided by X is equal to 1, and we're using this theorem to help us evaluate the following limit of the limit as x approaches 0 of of 39 x divided by 9 X. So we're using our theorem to help us evaluate the specific limit. So with that said, now that we know that we're ultimately trying to evaluate this limit, let us read off our multiple choice answers to see what our final answer might be. A is -3, B is 13 divided by 3, C is 13, and D is 58 divided by 3. Awesome. So our first step in order to solve this particular problem is we need to note that the expression sign of 39 x divided by 9 X has a sign function and let us note that the, uh, so the argument of the sign function is going to be 39 X. And because our expression has a sign function with an argument of 39 X, we will need to relate it to the known limit, and the known limit is we're going to be taking the limit, as X approaches 0 of sign of X divided by X is equal to 1. Fantastic. So now, our next step that we need to take is we need to factor out the constant, 1 divided by 9, from the expression. So, and let's write that in a separate color to keep things straight. So we're factoring out 19th, so we need to factor out this constant of 19th from the function. Or rather from the expression, I should say, not the function. So we're factoring out a constant value of 19th from the expression. So let me do that, we will find that sign of 39 x divided by 9 X is equal to 19th multiplied by of 39 x divided by X. And now our third step that we need to take is we need to apply the known limit. So in order to apply the known limit, we need to state that we need to let the value of U equal 39 X, and we need to note. That as X approaches 0, and this will be so as X approaches 0, you will equal 39 x, and we also note that this will be true as you approaches 0, and we should note that from U equals 39 X, it will also follow that X will equal you divided by 39. Awesome. So now our 4th step that we need to take is that from expression 3, or rather sorry I should say so focusing on our 4th step now, from step 3, we will take the expression, and we're taking the expression sign of 39 x divided by X can now be rewritten as follows. Sign of 39 x divided by X. Equals sign of you divided by divided by 39, which is equal to 39 multiplied by sign of you divided by you. And now our 5th step that we need to take is we need to recall and use the theorem, which once again, the theorem states that the limit as X approaches 0 of sine of X, divided by X is all equal to 1, which is essentially the same with So this is the same with the limit as you approaches 0 of sign of you divided by you is equal to 1. If it is in terms of you, and we can now substitute into our expression, and when we do that, we will find that the limit as X approaches 0 of sign of 39 x divided by 9 X. is equal to the limit, as X approaches 0 of 19th, multiplied by sign of 39 X. So sign of 39 X divided by X Awesome, which will be equal to the limit as you approaches 0 of 19th multiplied by 39, multiplied by Sign of you divided by you. Which will be all equal to 39 divided by 9, multiplied by the limit as you approaches 0 of sign of you divided by you, which will be all equal to. So when you plug in a value of 0 in for you because you is approaching 0, so when you plug in this value of 0 in for you, you will find that it will be equal to 39. 9 or 39 divided by 9 multiplied by 1, but we could simplify this expression, and you could use the fraction button on your calculator to find the simplified expression, or you could do it by hand, but ultimately, when you perform the simplification, your final answer should be 13/3 or 13 divided by 3, and that is our final answer. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be multiple choice answer letter B, which once again is 13 divided by 3 or 13/3s. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Use Theorem 3.10 to evaluate the following limits.
lim x🠂0 (sin 7x) / 3x

Key Concepts

Theorem 3.10 (Limit of Sin Function)

Limit Evaluation

Indeterminate Forms

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