Use l’Hôpital’s rule to find the limits in Exercises 7–52.
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Question

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

12. lim (x → ∞) (x - 8x²) / (12x² + 5x)

Accepted Answer

Welcome back, everyone. Evaluate the limit using Lapito's rule. Limit as X approaches infinity of 3 X2 minus X, divided by 5 X2 + 2. Let's begin this problem by rewriting the limit, limit as X approaches infinity. Of 3 x 2 minus X Divided by 5 X2 + 2. Remember that first of all, we begin with direct substitutions. We can just say X equals infinity in brackets, right? Because this is technically not correct, but it helps us understand what's happening. So we get 3 multiplied by infinity squared, which is basically positive infinity. Minus X, which is minus infinity, divided by 5 multiplied by infinity squared, which is positive infinity plus 2 it basically leaves us with positive infinity. So now we cannot really tell if we get infinity in the numerator. It's a difference of two infinities. So what we can do is simply perform factorization in the numerator. We're going to get X multiplied by 3 X minus 1, right? And this is going to be divided by 5 X2 + 2. And now we're going to perform the same step. So we're going to get infinity. Multiplied by 3 multiplied by infinity minus 1, which gives us infinity. So now this is infinity squared which is positive infinity, right? Meaning infinity minus infinity actually reduces to infinity in the numerator. We have just proved it using the factorization method. And now that we have shown that this indeterminate form is infinity divided by infinity, only now we can apply Lato's rule. Previously, we had a different indeterminate form, infinity minus infinity in the numerator, so this is inconclusive yet. But now that we can clearly see that this is infinity divided by infinity, we can proceed with the rule. The rule says that we want to take the derivative of the numerator and the derivative of the denominator. What we're going to do is apply the power rule. So, the derivative of 3 X squared is going to be 6 X and the derivative of negative X is -1. In the denominator, we're going to take the derivative of 5 X2, which is 10 X, and the derivative of 2 is 0 because 2 is a constant. Once again we're performing direct substitution, sex multiplied by infinity minus 1 is infinity. Somem multiplied by infinity is infinity, so still the same in the terminal form, meaning we're going to apply Lapito's rule once again. We're going to differentiate 6 X minus 1 and also differentiate 10 X, right? We're going to apply. The power rule, the derivative of X is 1 in each case, the derivative of -1 is 0. So we get 6 multiplied by 1 minus 0, that's 6. In the numerator And some in the denominator because the derivative of X is 1. And our 6 divided by 10 can be rewritten as. 3 divided by 5, right? So now we're taking the limit of a constant, which is equal to that constant itself. We don't have any access, right? So the answer to this problem is 3/5. Thank you for watching.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.12. lim (x → ∞) (x - 8x²) / (12x² + 5x)

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Use l’Hôpital’s rule to find the limits in Exercises 7–52.

12. lim (x → ∞) (x - 8x²) / (12x² + 5x)