Determine the following limits.

lim x→−...

Question

Determine the following limits.

lim x→−∞ (x+ √x^2−5x)

Accepted Answer

Welcome back, everyone. Evaluate the limit as X approaches negative infinity of the function F of X equals X + square root of X2 + 9 X. There are 4 answer choices. A 9 divided by 2, B 9 divided by 2, C-9, and D negative infinity. So let's define the limit. Limits X approaches negative infinity of X plus square root of. X2 + 9 X. If we apply direct substitution, while we get negative infinity plus square root of X2, that'd be infinity, plus 9 X that'd be negative. Infinity, so we get square root of infinity minus infinity, and this is an indeterminate form. We have to recognize that this function has a radical term, and then another term is added to it, so we can essentially apply the conjugate rule. What we're going to do is just rewrite it as limit as x approaches negative infinity, and now we want to multiply that function by its conjugate and then divide by the conjugate to make sure that it's unchanged. So we can just rewrite the function. X plus square root of X2 + 9 X, and we're going to multiply it by its conjugate which is X minus square root of. X2 + 9 x and now let's divide it by the conjugate to make sure that it's unchanged x minus square root of x2 + 9 X. Now why is this useful? Well, we can see that we have a factored form of the difference of two squares, where A is X and B is the radical term. So we end up with A + B multiplied by a minus B. Which in a standard form is A2 minus B2. So we get X2 minus. Now, when we square the radical term, it is going to be X2 + 9 X. And for the denominator, let's leave it as it is x minus square root of. X2 + 9 X. Let's simplify. We get limit as X approaches negative infinity of -9 X, so we can factor out the negative sign. Divided by X minus and we can factor out X2 from the radical, right? When we factor it out and when we take the square root of X2, that's going to give us the absolute value of X, right, because square root of X2 is the absolute value of X. And within the radical, we end up with 1 because X2 divided by X2 gives us 1 plus 9 x divided by X2, which is 9. Divided by x. Now if we apply the right substitution, well, we are going to get infinity divided by some sort of the differences of infinities, right? So essentially once again we get an indeterminate form, but we notice that this is a rational function, meaning we want to divide both sides by X with the highest exponent, and that's X to the power of first. So if we divide each term by X, we're going to get 9 in the numerator. In the denominator, we're going to get X divided by X, which is 1. Then we get the absolute value of x divided by. X And now for the absolute value since X is going to be negative. We have to recall that the absolute value of X is simply negative X, when X is negative. So now let's rewrite the limit as limit when X approaches negative infinity. In the numerator, we have 9, in the denominator, we have 1 minus, and now the absolute value of X is going to become negative X because X is negative. So we get negative X divided by X, which is -1. And because they're seeing negative sign in front. We get minus -1, which is basically 1. And then let's rewrite the square root term 1 + 9 divided by X. As x approaches negative infinity or infinity, any term in the form of a number divided by X approaches 0. So 9 divided by X approaches 0. And now we can apply direct substitution once again, so we get negative. 9 divided by 1 plus. Squirrel it off. 1 + 0. And that's negative 9 divided by 2. Which is a finite number and this means that we have successfully evaluated the answer, which is option B, -9 halves. Thank you for watching.

Determine the following limits.

lim x→−∞ (x+ √x^2−5x)

Key Concepts

Limits

Square Root Function

Dominant Terms in Polynomials

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