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)=√x^2+2x+6−3 / x−1

Accepted Answer

Welcome back, everyone. Determine the horizontal asymptotes of the function G of X is equal to square root of X2 + 4 X + A divided by X + 1, and we're given 4 answer choices A plus Y equals -1 and 1 B 2 and 2, C-4 and 4, and D-5 and 5. So if we want to identify the horizontal asymptos, we want to identify two limits. The first one is limit as X approaches positive infinity of the function, which is a rational function in the form of square root of X2 + 4 X + 8. Divided by x + 1. If we substitute infinity, well, we're going to get an indeterminate form which is infinity divided by infinity, and since x approaches infinity, we can essentially identify the limit if we divide all of our terms by X with the highest exponent in the denominator, and that's x to the power of first. But we have to be extra careful because we have square root in the numerator, right? So what we're going to do first is factor out. The X squared. Under the radicals, we're going to get X2 in 1 + 4 divided by X plus 8 divided by X2. And now for the denominator, let's just keep it as X. Plus one for now. Now let's take square root of X squared. We have to recall that this is equal to the absolute value of X, not just X, right, because we can have two solutions, positive and negative, which leaves us with 1 + 4 divided by X. Plus 8 divided by x2. And now in the denominator we have X plus 1. Now we can divide all of our terms by X because this is. The highest exponent. Of the independent variable in the denominator. So we're going to get limit as x approaches infinity of the absolute value of x divided by X multiplied by square root of 1 + 4 divided by X plus 8 divided by x2d. And now in the denominator we're going to get. 1 plus 1 divided by X. And now if X approaches infinity, well, we can essentially say that X is greater than 0 in this case, so we can drop the absolute value. And we can now see that x divided by X simply gives us a value of 1. Now, as X approaches infinity, the fractional terms are going to approach 0 because we're dividing a number by infinity. And overall we can see that this gives us a finite limit, right? square root of one. Divided by 1, which is equal to 1, and this is our first asymptote. Now for the second one, we want to evaluate that same limit as x approaches negative infinity. And we're going to use the simplified form, but now we have to understand that X is less than 0, and this is going to give us. Negative X divided by X if we decide to drop the absolute value, which is a piecewise function. Now we're going to rewrite the radical term square root of 1 +4 divided by x + 8 divided by x 2 in the denominator 1 + 1 divided by X. So now we can see that negative X divided by X gives us -1, and that's the only change. So we're going to get. -1 multiplied by square root of 1. And in the denominator, that's still one. And this is our second asymptote which is -1. So we have two asymptotes Y equals -1 and 1, which corresponds to the answer choice A. Thank you for watching.

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

f(x)=√x^2+2x+6−3 / x−1

Key Concepts

Limits at Infinity

Horizontal Asymptotes

Rational Functions

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