Determine the following limits.

lim x→∞...

Question

Determine the following limits.

lim x→∞ x^4+7 / x^5+x^2−x

Accepted Answer

Welcome back, everyone. Evaluate the limit as x approaches negative infinity of the function F of X equals x6 minus 2 X2 + 3 divided by 2x6 plus x minus 5, and we are given four answer choices A says 2, B 1/2, C3, and D 35. So we want to evaluate the limit. As x approaches the negative infinity of F of X, which is X6 minus 2 X2 + 3. Divided by 2 x 6 plus x minus 5, and if we apply the direct substitution well we notice that we get infinity minus infinity divided by infinity, and this is an indeterminate form. So what we have to do is notice that this is a rational function, right? So if we have a rational function that consists of two polynomials in the numerator and the denominator, we have to identify the independent variable with the highest exponent, and that's x to the power of 6 for both our numerator and the denominator. And from here, we have to divide each term. Of each polynomial by x to the power of 6, so we're going to get limit x. Approaching negative infinity. Now if we divide X to the power of 6 by x the power of 6, we can divide that x the power of 6 divided by x the power of 6. This is going to give us 1, right? And then we're going to perform the same step. Or every other term. We have our numerator and now for the denominator, we got 2 X 6 divided by X6 plus X divided by X6 minus 5. And now let's simplify. We get limit as X approaches negative infinity of 1 minus 2, and now for the second term. We have 2 in the numerator, X2 divided by X6 gives us X4. And then for 3, well, we also have to include 3 divided by X6, right? Because this was a free term. And that's what we have to do for 5 as well. So let's add that. Plus 3 3 divided by X6, and in the denominator, we get 2 plus 1 divided by X5 minus 5 divided by X6. Now if X approaches either negative infinity or positive infinity, any number divided by infinity is going to approach 0. So we can say that 2 divided by X4X approaches negative infinity is going to approach 0. Which is also applicable to 3 divided by XX. 1 divided by x 5th and 5 divided by X6. It doesn't matter if we have a positive or a negative sign in front of each fraction, right? Essentially any number divided by X where X approaches negative infinity or positive infinity is going to approach 0. Just from different sides, from the left or from the right, but this is irrelevant because we have two whole numbers, 1 and 2, right, for our limit. And essentially they define the limit, so we get one. Divided by 2, which is 1/2 for the final answer, right? And this is now a finite number. We don't have any indeterminate forms, so we can conclude that the correct answer to this problem is B12. Thank you for watching.

Determine the following limits.

lim x→∞ x^4+7 / x^5+x^2−x

Key Concepts

Limits at Infinity

Polynomial Functions

Dominant Terms

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