Determine the following limits.

lim x→∞...

Question

Determine the following limits.

lim x→∞ 6x²/(4x^2+√(16x⁴ + x²))

Accepted Answer

Welcome back, everyone. Evaluate the limit as x approaches infinity of the function F of X equals 24x4 divided by 8X4 plus square root of 25 x 8 plus x4, and we have 4 answer choices A 24 divided by 13, B 13 divided by 24, C3, and D 24. So let's define the limit. We have limit as X approaches infinity of F of X, and the FF X is 24X4 divided by 8 X4 plus square root. Of 25 8 + X4. First of all, we have to apply direct substitution, so let's substitute X equals infinity. 24 multiplied by infinity gives us infinity, and in the denominator we end up with 8 multiplied by X4th as x approaches infinity, that's basically infinity, plus square root of infinity, which is basically infinity. And this is an indeterminate form, which means that we have to understand that this is a rational function. So, as X approaches infinity or negative infinity, we can divide both sides of the rational function by X with the highest exponent. Let's go ahead and do that and we can notice that it's going to be X to the power of 4th because within the radical we have square root of X8, which is also X4th. So in the numerator, we get 24 X4 divided by X4. In the denominator. We get 8 x 4 divided by X4 plus square root of, and now because we have a radical that's going to be x to the power of eight. So we get 25 divided by X to the power of 8, and we have X to the power of eth in the numerator. And then plus x to the power of 4 divided by X the power of 8. Now, let's cancel out. X is the power of 4th. On both sides of each fraction. And the power of eth. Let's simplify the limit. We get limit as X approaches infinity. Of 24 divided by 8. Plus square root off. 25 + 1 divided by x to the power of 4th and now as x approaches infinity, the fractional term is going to approach 0 because we will have a number divided by an infinitely large number. Which means that now we can attempt direct substitution once again and this gives us 24 divided by 8. Plus square root of 5. Which is 24. Divided by 8 + 5, right? square of 25 is 5, so we don't need a radical sign. So we get 24 divided by 8 + 5, which is 13. And that's our final answer. That's option A. Thank you for watching.

Determine the following limits.

lim x→∞ 6x²/(4x^2+√(16x⁴ + x²))

Key Concepts

Limits at Infinity

Dominant Terms

Rational Functions

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