Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any ...

Question

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x) = (√(16x⁴ + 64x²) + x²) / (2x² − 4)

Accepted Answer

Welcome back everyone to another video. Evaluate the limit as x approaches positive and the negative infinity and identify any horizontal asymptotes for the function F of X is equal to square root of 36 x to the power of 4 + 108 x to the power of 2 + x2 divided by 3 x 2 minus 9. As the problem suggests, we want to evaluate two limits, let's valuate the first one, limit as X approaches infinity. We're going to substitute the given rational function. Let's introduce our radical term, and then we also have x2d as a free term, and this is divided by 3 x2 minus 9. If we use direct substitution, well, we're going to get infinity plus infinity divided by infinity, which is an indeterminate form. And since x approaches infinity well. We simply want to divide both sides of our fraction by X with the highest exponent in the denominator, and that's X squad. So let's rewrite the limit limits x approaches infinity. In the numerator we're going to divide everything by x2d since we have square root. Well, we need to divide by square root of x the power of 4th, so we're going to get 36 x to the power of 4th divided by X to the power of 4th plus 108 x 2 divided by. X to the power of 4th and then we also have X2d which is going to be divided by x2. In the denominator, we get 3 x 2 divided by X2 minus 9 divided by X2. So now our limit becomes limit as X approaches infinity. Of square root. Of 36 + 108 divided by X squared plus 1 divided by We minus 9 divided by X2. Now, as X approaches infinity, the fractional terms approach 0 because we have a number divided by infinity. And from here we can evaluate the result. Let's attempt the right substitution once again. We end up with screwed of 36 + 1 divided by 3, which gives us 7/3. So this is our first horizontal asymptote, Y equals 7/3. And now for the other horizontal asymptote we want to evaluate that same limit as x approaches negative infinity. We're going to have our radical term square roots of 36 + 108 divided by X squared. Plus one in the denominator. 3 minus 9 divided by x2. Now it doesn't really matter if x approaches positive or negative infinity because X2 will approach positive infinity in in each case. So the fractional terms still approach zero, and that means we end up with the same value of the limit 7/3. So Y equals 73 is the only horizontal asymptote. And that's our final answer. Thank you for watching.

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.
f(x) = (√(16x⁴ + 64x²) + x²) / (2x² − 4)

Key Concepts

Limits at Infinity

Horizontal Asymptotes

Rational Functions

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