60–81. Limits Evaluate the following limits. Use l’Hôpital’s R...

Question

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.

lim_Θ→0 (3 sin² 2Θ) / Θ²

Accepted Answer

Welcome back, everyone. Evaluate the limit. Use Lato's rule if necessary. Limit SX approaches 0 of 5 squared of 4 x divided by 8 x 2. We're given 4 answer choices AS 160, B 0, C 10, and D does not exist. So let's rewrite the given limit. Limit as X approaches 0. We have 5. Sine squared of 4 x divided by 8 x 2. We always begin with direct substitutions, we get 5 of 0, which is 0. And we divide that by 8 multiplied by 0, which is also 0, and this is an indeterminate form. And this is the indeterminate form which allows us to use Lapiel's rule. Lapidel's rule says that if we have 0 divided by 0 or infinity divided by infinity, then we can take the derivative of the numerator. And the derivative of the denominators we are going to differentiate 5 squared of 4 X and 8 X squared separately. Let's valuate each derivative. We're going to get the limit as x approaches 0. We can factor out the constant, which is 5. The derivative of sin squared is going to be 2 sin of x using the power rule, right? That's because we bring down the 2 and decrease the original exponent by 1 unit. But we have an inner function which is sin of 4 X, so we also multiply by the derivative of sine which is cosine. And another inner function is 4 X. Right, so the derivative of 4 X is 4, meaning we have applied the chain rules twice since we have, we had several inner functions. This is divided by the derivative of 8X2d, which is 16 X. That's because the derivative of X2 is 2 X. From here, we can just simplify. We're going to get limit as X approaches 0. Let's understand that 24 X cosine 4 X. Can be rewritten as sign. Of 8Xs based on trigonometric identities, and this gives us 20. Sign of 8 X in the numerator. And our denominator now has 16 acts. Now let's apply the direct substitution once again. We're going to get 20 multiplied by sign of 0, which is 0. In the denominator 16 multiplied by 0 is 0. This is still the same indeterminate form, which means that once again we're going to apply Lapito's rule. Which gives us limit as x approaches 0 off. 20 multiplied by the derivative of sine of 8 X. Divided by 16 The derivative of X. Let's evaluate each derivative. We're going to get limit as x approaches 0. In the numerator we get 20 multiplied by the derivative of sine, which is cosine. Of the same angle 8 X. And once again we have an inner function, so let's apply the chain rule which says that we're applying. The rule in such a way that we're multiplying cosine of 8X by the derivative of the inner function, so the derivative of 8X. We know that the derivative of 8 X is equal to 8 because the derivative of X is 1. And now for the denominator, the derivative of X is 1, so we simply get 16. Now we're attempting the right substitution once again. So overall, that's 20. Multiplied by cosine of 0, which is 1, multiplied by 8. And in the denominator, we have 16. This is 160 divided by 16, which is 10. It is a finite limit, meaning we have successfully determined the final answer. The only answer to this problem corresponds to the answer choice C sum. Thank you for watching.

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_Θ→0 (3 sin² 2Θ) / Θ²

Key Concepts

Limits

L'Hôpital's Rule

Trigonometric Functions

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