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_x→1 (x⁴ - x³ - 3x² + 5x -2) / x³ + x² - 5x + 3

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. Evaluate the limit. Use Lapital's rule if necessary. The limit as X approaches 1 of 7 x 5 plus 5 x 4 minus 3 x 3 minus 6 X 2 + 8 X minus 11. All divided by 3 x 4 minus X 3 plus 2 X 2 minus 9 X + 5. Awesome. So it appears for this particular problem, we're ultimately trying to evaluate this limit. So we're trying to figure out what this limit is equal to. So we're trying to evaluate our limit. Awesome. So with that said, let's read off our multiple choice answers to see what our final answer might be. A is 31 divided by 2, B is 21 divided by 2. C is 45 divided by 4, and D is negative 21 divided by 2. Awesome. So first off, let us try the direct substitution method first to see if we can evaluate our limit that way. So we need to take the limit as X approaches 1, and using the direct substitution method, we just plug in a value of 1 in for X. So we need to take 7 multiplied by 1 to the power of 5, plus 5 multiplied by 1 to the power of 4 minus 3. Multiplied by 1 to the power of 3, because once again we're plugging in a value of 1 in for X minus 6, so 6 multiplied by 1 to 2, plus 8 multiplied by 1. -11. All divided by 3 multiplied by 14 minus 1 of 3. Plus. 2 multiplied by 12 minus 9 multiplied by 1. Plus 5. So when we plug in that big old long expression into our calculator, we will find that the limit as x approaches 1 using the direct substitution method will give us an indeterminate form as we get 0 divided by 0, and because it gives us an indeterminate form, that means we have to use Lahapatal's rule in order to solve for this specific limit. So in order to evaluate this limit, we have to use Lahapatal's rule. So we're calling and using Lahapathal's rule. We need to take the limit. As X approaches 1 of 7 X 5 plus 5 x 4 minus 3 X 3 minus 6 X 2 + 8 X minus 11. Fantastic. And then we need to take this, and we need to divide it by 3 X 4 minus X 3 plus 2 X 2 minus 9 X + 5. Fantastic. And what we need to do now is now we need to take the derivative with respect to X, so we need to take that this is equal to the limit. So essentially, so I don't have to write it again, we need to take the we need to take the derivative with respect to X for the numerator and the denominator. So essentially, we need to take D by DX, which means it's the fancy way of saying the derivative with respect to X of the numerator. And we also need to take the derivative of the denominator. Which is the bottom, so numerators at the top, denominators at the bottom, so when you take D by DX of the numerator and denominator. So when we take the derivative and note that we're taking the first derivative, we will find that this is equal to The limit As, and once again we're take when we take the derivative of the numerator and the denominator, we will find the limit as X approaches one of. 35 X to the power of 4. Plus 2. X to the power of 3. 9 X to the power of 2, minus 12 X. Plus 8, all divided by 12. X 3 minus 3 x 2, plus 4 X minus 9. And now what we need to do is we now can try our direct substitution method now that we've took the first derivative of the numerator and the denominator. So when we do that, we will find the limit as X approaches 1 of 35 multiplied by 1 to the power of 4, because once again we're plugging in a value of 1 in for our values of X. So 35 multiplied by 1 to the power of 4. Plus 20, and then we need to take 20 multiplied by 1 to the power of 3. And then we need to take -9 multiplied by 1 to 2 minus 12 multiplied by 1 + 8. All divided by 12, multiplied by 1 to the power of 3, minus 3 multiplied by the multiplied by 1 to the power of 2, plus 4 multiplied by 1. -9. And when we plug that expression into our calculator, we will find that our final answer infraction form is 21 divided by 2. And that is our final answer. Hooray, we did it. So looking at our multiple choice answers, the final answer has to be multiple choice answer, letter B, which states once again that our final answer is 21 divided by 2. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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

lim_x→1 (x⁴ - x³ - 3x² + 5x -2) / x³ + x² - 5x + 3

Key Concepts

Limits

L'Hôpital's Rule

Polynomial Functions

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