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→∞ (3x⁴ - x²) / (6x⁴ + 12)

Accepted Answer

Evaluate the limit and use the law's rule if necessary, where we have the limit as X approaches infinity of 10 x 4th minus 3 X 2 divided by 8 X 4th plus 16. We also have 4 possible answers, being negative 4/50, 0, 5/4, and 8/3. Now, to solve this, let's first see what happens with our limit. We have limit as X approaches infinity. If we were to plug in this infinity, We get 10 multiplied by infinity to the 4th, minus 3 multiplied by infinity to the 2nd. Divided by 8. Infinity to the 4th plus 16. This gives us the form of infinity divided by infinity. Now, when you have an indeterminate form, such as infinity divided by infinity, we can make use of Loppital's rule. This rule states that if we have an indeterminate form, such as infinity divided by infinity, Our limit Can be found by taking. The derivative of our numerator and our denominator. So let's take the derivative of the numerator and the derivative of the denominator. We have 10 x 4 minus 3X 2 divided by 8 X to the 4th plus 16. By Lapital's rule, this turns into 40 X3. Minus 6 X divided by 32. X To the 3rd This form, once again, is infinity divided by infinity, because if we plug in an infinity, We would get infinity minus infinity on the numerator and infinity and the denominator. So we can use Lapital's rule once more. We'll get 120 X to the second. M 6 Divided by 96 X to the 2nd. Now, we plug in our value again. Of infinity, we get another form of infinity divided by infinity. So we will take one more derivative. This gives us 240 X. Divided by 192 X. Now, we notice that there's an X in a numerator and an X in the denominator that can be canceled out. We end up getting 240 divided by 192. Where X approaches infinity. Now, since there's nowhere for the infinity to go, we can just simplify this fraction to find our limit. 240 divided by 192 simplifies to be 5 divided by 4. So our limit 54ths corresponds. With answer C. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→∞ (3x⁴ - x²) / (6x⁴ + 12)

Key Concepts

Limits

l'Hôpital's Rule

Polynomial Functions

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