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→ ∞ (4x³ - 2x² + 6) / (πx³ + 4)

Accepted Answer

Below there, today we're going to 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 Lahapatal's rule if necessary. The limit as x approaches infinity of 9 x 3 minus X2 + 3, all divided by 2 multiplied by pi multiplied by X 3 + 5. Awesome. So it appears for this particular problem. We're asked to take this limit that is provided to us, and we're asked to evaluate it. So we're ultimately trying to figure out what the what the evaluated limit is, and that will be our final answer we're trying to solve for. So with that said, let's read off our multiple choice answers to see what our final answer might be. A is 9 divided by 2 multiplied by pi. B is 15 divided by 2 multiplied by pi, C is 9 divided by pi, and D is 27 divided by pi. Awesome. So first off, we need to recall a note that since X approaches infinity, both the numerator and denominator will approach infinity, leading to an indeterminate form. And what does that mean? An indeterminate form means ultimately when we evaluate this limit using direct substitution, so if we plug in a value of infinity in for X, we will get an indeterminate answer, which will be infinity divided by infinity. Awesome. So once again, that's when you plug in a value of infinity in for X, you'll get like infinity multiplied by infinity, divided by infinity, which once again will just give you an indeterminate form, which is infinity divided by infinity. And because this limit is and produces an indeterminate form, we can therefore recall and apply Lajapatal's rule. And as we should recall, Lajapatal's rule states that if the limit, which will say limit as X, approaches A of F of X divided by G of X is in the form. 0 divided by 0, which is 1 indeterminate form, or infinity divided by infinity, then we could say that it could be evaluated as And let's move this to the side, so we could say that this is evaluated. As a once again, that's a note be indeterminate form, we could say that this limit as X approaches A F of X, divided by G of X is going to be equal to the limit as X approaches. A of F of X divided by G of X. So what does this mean? So this prime symbol, it means that we're taking the first derivative. So basically it's saying that this limit as X approaches A of F of X divided by G of X is equal to the Limit as X approaches A of the first derivative of F of X, divided by the first derivative of G of X. And this is due to the fact that provided the limit is the limit on the right exists, so provided due to the fact that the limit on the right exists. So essentially, we need to find the the derivative of the numerator, which is the value or the expression on top, and then we need to find the derivative of the denominator, which is the Bottom expression. Or or the bottom of the division bar expression. So, with that said, let's find the derivative of the numerator first, which when you take D by DX, which D by DX is just a fancy way of saying we're taking the derivative with respect to X. Of 9 X to the power of 3 minus X2 + 3, which when we perform that derivative, we will get 27 multiplied by X to the power of 2 or 27 X to the power of 2 minus 2 X, fantastic. And similarly for the denominator, so we take D by DX of 2 multiplied by pi, multiplied by X to the power of 3. Plus 5, and when we perform this derivative, we'll get 6 multiplied by pi, multiplied by X2. Fantastic. So now our next step that we need to take is we now can plug in our now since we took the derivative, we can plug in our numerator and denominator into our expression, and we will get the following. We will get that limit as X approaches infinity of 27. X to the power of 2 minus 2 X divided by 6 multiplied by pi, multiplied by X to the power of 2. So once again, this produces an indeterminate form because right now if you plug in a value of infinity using direct substitution in for X, we will get an indeterminate form. So right now this equal, we'll make a note that this gives us an indeterminate form of infinity divided by infinity. So that means we need to apply Lahapatal's rule one more time. So we need to take the derivative of the numerator and denominator again. So when we do that, And let's put this in parentheses. So this is a little, this is a side note. So now, when we take the derivative of the numerator. Once again, we will get, once again, we're taking the first derivatives, so I'll make a note of this. So if we take D by DX D by DX for the top and bottom. We will get, as a result, we take the the derivative of the numerator, we'll get 54 X minus 2 divided by 12 multiplied by pi, multiplied by X. And once again, as you could see, when we plug in a value of infinity in for X, We will get an indeterminate form again, so that means, because once again if we plug in infinity in for X, this will give us an indeterminate form yet again. So we have to do Lahapa's rule against we need to take the derivative of the numerator and denominator one last time, and when we do, we will find that it's equal to 54 divided by 12 multiplied by pi. So, when we simplify this expression, we will find when we use direct substitution means there's no value of X, and so we simplify. We will find that our final answer must be equal to 9 divided by 2 pi. And that's it. That is our final answer. Hooray, we did it. So looking at our multiple choice answers the limit has to be multiple choice answer letter A, which once again is 9 divided by 2 pi or 9 divided by 2 multiplied by pi. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ ∞ (4x³ - 2x² + 6) / (πx³ + 4)

Key Concepts

Limits

l'Hôpital's Rule

Polynomial Functions

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