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) = (x² − 4x + 3) / (x − 1)

Accepted Answer

Welcome back, everyone, to another video. Evaluate the limits as x approaches infinity and thus x approaches negative infinity and identify any horizontal asymptotes for the function F of X is equal to X2 minus 13 x + 40 divided by X minus 5. So the first limit that we want to consider is limit as x approaches negative infinity of the given function, which is X2 minus 13 X. Plus 40 divided by x minus 5. If we apply the right substitution, well, we are going to get infinity plus infinity divided by. Negative infinity, right? And this is an indeterminate form. So what we want to do is simply understand that this is a rational function. So we're going to divide both sides of that function by The independent variable with the highest exponent in the denominator. The reason why we're doing this is because X approaches negative infinity. So let's go ahead and divide all of our terms by X. We're going to get X2 divided by X minus 13 x divided by X plus 40 divided by X. And now in the denominator we're going to get X divided by X minus 5 divided by X. Now, let's evaluate The expression let's apply the rules of exponents. In the numerator, we're going to get X minus 13 + 40 divided by X. In the denominator, we're going to get 1 minus 5 divided by X. Now 5 divided by X, X approach is negative infinity, approach is 0. In the numerator 40 divided by X also approaches 0 because we have a number divided by an infinitely large number, which is also negative. And now from here we can Evaluate the result in the numerator, and we're going to get negative infinity minus 13, which is negative infinity. In the denominator we have 1, so negative infinity divided by 1 gives us negative infinity. This is not a finite number, so we don't have any horizontal asymptotes yet. But we also need to evaluate that same limit as x approaches positive infinity, and we're going to use the simplified form X minus 13 + 40 divided by X. And in the denominator, we have 1 minus 5 X. So the whole difference is that now we are dividing by. Positive infinity and we end up with infinity minus 13, which gives us infinity and in the denominator we simply have 1. So now we end up with. A limit of positive infinity. So we have evaluated the two limits, and none of them is a finite number, meaning there are no horizontal asymptotes. And we have finished the problem, so let's label our answers negative infinity for the limit as x approaches negative infinity, positive infinity as X approaches positive infinity, and we also know that there are no horizontal asymptotes. Thank you for watching.

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.
f(x) = (x² − 4x + 3) / (x − 1)

Key Concepts

Limits at Infinity

Horizontal Asymptotes

Polynomial Division

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