17–83. Limits Evaluate the following limits. Use l’Hôpital’s R...

Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→2π (x sin x + x² - 4π²) / (x - 2π)

Accepted Answer

Welcome back, everyone, to another video. Evaluate the given limit using Lapital's rule if applicable. Limit as X approaches pi of X2 cosine of X + 3 X + 2 minus 3 pi all divided by X minus pi. So let's rewrite the limit. The limit as x approaches pi of X2 cosine of x. 3 x + 5 squad minus 3 pi. All of that is divided by x minus pi. When we begin, we simply want to apply the right substitution, meaning for every X we're going to substitute pi. This means that we're going to get pi squared. Cosign of pie. Plus 3 I. Plus 5 squad minus 3 pi and this is divided by pi minus pi. Now if we simplify, we're going to get 0 in the numerator. And 0 in the denominator. And this is an indeterminant form, which is perfectly fine for Lapito's rule, meaning whenever we get 0 divided by 0 or infinity divided by infinity, we can go ahead and apply Lapito's rule. Now Laptal's rule says that what we want to do for this indeterminate form is simply differentiate the numerator and the denominator. So let's begin with the denominator. Now negative pi is a constant and the derivative of X is 1, so we simply get 1 in the denominator. For the numerator, we have 12 minus 3 pi. Not that these are constants, so their derivatives are 0, and this means that we want to identify the derivative of X2 cosine x + 3 X. So let's identify the derivative. First of all, we have a product, X2 cosine x. According to the product rule, we're taking the derivative of X2, multiplying by cosine of X, and adding X2, which is multiplied by the derivative of cosine of X. So here we have applied the product rule, and then the derivative of 3 X is simply 3 because the derivative of X is 1. So let's identify the missing derivatives. We get 2 X. Cosine of X, the derivative of cosine is negative sign, so we simply get negative X squared sin of x. And then finally we get 3, so that's our numerator. We can simply rewrite it. 2 X. Cosine x. Minus X2 x x. 3 And from here, once again, we are attempting direct substitution. So X approaches by this means that we get 2. I Cosine of pi. Minus I squared Sign of pie Plus 3. All of that is divided by 1, so our denominator doesn't make any difference. We have to recall that sign of pi is equals to 0, so our middle term is simply 0 and we get 3. Now, cosine of 5 is equal to. -1 So we get 2 pi multiplied by -1, which gives us -2 pi. So the final answer is 3 minus 2 pi. This is now a finite number, not in indeterminant form, meaning we have arrived at the final answer to this problem. Well done. Thank you for watching.

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→2π (x sin x + x² - 4π²) / (x - 2π)

Key Concepts

Limits

L'Hôpital's Rule

Continuity

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