Finding Limits of Differences When x → ±∞
...

Question

Finding Limits of Differences When x → ±∞

Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)

lim x → ∞ (√(x² + x) − √(x² − x))

Accepted Answer

Welcome back, everyone. Determine the limit of the given function using its conjugate. Lim SX approaches infinity of square root of X2 + 2X minus square root of X2 minus 2X. A says 0, B1, C2 and D does not exist. So let's write down the limit. Limits X approaches infinity. Of square root of X2 + 2 X. Minus square root of X2 minus 2 X. What we're going to do is simply use the method given to us in the problem. It says that we are going to use the conjugate of this function, which means that we are multiplying and dividing by our function with an opposite sign between the two radicals. So let's multiplied by a square root of X2 + 2 X. And now instead of the negative sign, we're going to add the positive sign. Followed by square root of X2 minus 2 X, our next radical term. And what we have to do is basically divide by the same conjugate. Square root of X2 + 2 x +. Square root of X2 minus 2 X. That's because we want to ensure that our original function is unchanged. What we can do now is simply multiply the terms in the numerator. And as we can see, we have the factor form of the difference of squares. So we have to square our first term. Which gives us X2 + 2 X. And subtract the square. Of the second term, which is x2 minus 2 X. We're not going to change our denominator. It remains the same. And now for the numerator, we can combine the like terms. By also distributing the negative sign, so X2 minus X2 is going to give us 0 and 2 X minus -2 X gives us 4 X. Considering the denominator, we can just rewrite that. And now if we Attempt direct substitution. Which we technically cannot do because X approaches infinity, right? We are going to notice that. We're getting an infinitely large number in both the numerator and the denominator, right, so infinity divided by infinity. Is an indeterminate form. But because we have the irrational expression, we can divide all of our terms by X. So let's begin with the numerator. We get the limit as X approaches infinity where X divided by axis 4. And considering the denominator, because we have radical terms, we're going to divide by square root of X2. Which is exactly X because X is positive where approaching positive infinity. So our radical terms become square root of X2 divided by X2, which is 1. Plus 2 x divided by X word, which is 2 divided by X. And our next radical term is going to be one. Minus 2 divided by X. Now, the fractional terms are going to approach 0 because a non-zero number divided by an infinitely large number gives us a limit of 0. And this means that the limit is 4 divided by square root of 1 plus square root of 1, which is 4 divided by 2. And that's simply 2. So our final answer corresponds to the answer choice C. Thank you for watching.

Finding Limits of Differences When x → ±∞

Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)

lim x → ∞ (√(x² + x) − √(x² − x))

Key Concepts

Limits at Infinity

Conjugate Multiplication

Simplifying Radical Expressions

Watch next