Determine the following limits.
lim x→∞ (x⁴ ...

Question

Determine the following limits.

lim x→∞ (x⁴ − 1) / (x⁵ + 2)

Accepted Answer

Welcome back, everyone. Evaluate the limit. Limit as x approaches infinity of X to the power of 6 minus 7 divided by X to the power of 7 + 11. We're given 4 answer choices A says 0, B 7 divided by 11, C 1, and D 7. So let's rewrite the limit, limit as x approaches infinity of X to the power of 6 minus 7 divided by X to the power of 7 + 11. If we apply the right substitution, well, we are going to get infinity, right, because X to the power of 6 is infinity to the power of 6, that's infinity. -7 has no effect and in the denominator we're getting infinity as well and this is an indeterminate form. Now whenever we get this in the terminate form and X approaches infinity, if we have a rational function, we can divide both sides of that function by the variable with the highest exponent, and that's x to the power of 7. So let's go ahead and do that. Limit as x approaches infinity. And now we're going to divide both sides of that fraction by X to the power of 7th. So we get X to the power of 6 divided by X the power of 7th, minus 7 divided by X the power of 7th, and in the denominator, we're going to get X. To the power of 7th divided by X to the power of 7th. Plus 11 divided by x the power of 7th and now let's simplify. The limit becomes limit as x approaches infinity of 1 divided by x is the power of 7th minus 7 divided by x is the power of 7th, and in the denominator we get 1 plus 11 divided by X is the power of 7th. Now each fractional term approaches 0 because x approaches infinity, right? So a number divided by infinity approaches 0. So we can say that in the denominator we have 0 minus 0. That's our numerator, right? And in the denominator we have 11 divided by X is the power of 7th. This term also approaches 0 because x approaches infinity. And now, once again, if we apply the right substitution, we're going to get 0 divided by 1 + 0, which is 1, and that's 0. This is now a finite number and essentially we have evaluated the limit, right? Because that's a finite number and therefore, the correct answer choice to this problem corresponds to the answer choice A. Thank you for watching.

Determine the following limits.
lim x→∞ (x⁴ − 1) / (x⁵ + 2)

Key Concepts

Limits at Infinity

Polynomial Functions

Dominant Term

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