Evaluate the following limits. Use l’Hôpital’s Rule when it is...

Question

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

lim_x→ -1 (x⁴ + x³ + 2x + 2) / (x + 1)

Accepted Answer

Welcome back, everyone. Evaluate the limit. Use Lapito's rule if necessary. Limit as X approaches -4 of X to the power of 4 + 4 X cubed + 6 X + 24 divided by X + 4. We're given 4 answer choices AS 58, B 36, C 24, and D 38. So let's go ahead and first we'll apply the right substitution. We have our limit. Let's rewrite the function X to the 4th plus 4 x cubed + 6 X + 24 divided by. X + 4. What we want to do is simply substitute x equals -4 for every term. We get -4 to the power of 4th plus 4 multiplied by -4, raise to the power of 3, plus 6 multiplied by -4 + 24, and we're dividing that by -4 + 4. This leads to an indeterminate form, 0 divided by 0, right? We have 0 in the numerator and the denominator because this is an indeterminate form we can apply the Lapatos rule for 0 divided by 0 or infinity divided by infinity. Which means that we have. To take the derivative of the numerator, so we're differentiating. Our numerator, let's simply rewrite it and apply the derivative. And now in the denominator, we also want to take the derivative. This is what the Lapitos rule says, right? If we get 0 divided by 0 infinity divided by infinity, we can simply differentiate the numerator and the denominator and then apply the right substitution once again. So in the numerator we're going to get 4 X cubes applying the power rule. La We multiplied by 4, that's 12 x 2 for our second derivative. The derivative of 6 X is 6, the derivative of 24 is 0, right? Because it's a constant. Now, X + 4 gives us a derivative of 1, right? So essentially, we now want to apply direct substitution once again and because we don't have any denominator, we can simply multiply 4 by 4 cubed plus 12 multiplied by -4 squared, and then we are adding 6. And this is not going to be an indeterminate form, right? We don't have a denominator. It's basically one. So it looks like we will get a finite number. So we get 4 multiplied by -64, that's -256. We are then adding 12 multiplied by 16. Which is 192, and then we are adding 6. So now, if we combine those, well, we are going to get -58, which is our final answer, and that corresponds to the answer choice A. Thank you for watching.

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

lim_x→ -1 (x⁴ + x³ + 2x + 2) / (x + 1)

Key Concepts

Limits

l'Hôpital's Rule

Polynomial Functions

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