Finding Limits of Differences When x → ±∞
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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 + 3x) − √(x² − 2x))

Accepted Answer

Welcome back, everyone. Find the limit of the given function using its conjugate. A says 0, B 3 divided by 4, C1 and D 7 divided by 2. We're given the limit as X approaches infinity. Of square root. Of X2 + 4 X minus square root of X2 minus 3 X. What we're going to do is apply the concept of the multiplication by the conjugates, so we're going to get limit as X approaches infinity. We're going to write the fraction sign. In the numerator, we're going to rewrite the original function. So we're going to write the two radical terms. And now our goal is to multiply by the same term. With the positive sign between the radicals, that's the conjugate of the original function. So we get square root of X2 + 4 X plus. Square root of x2 minus 3 X. Because we want to ensure that our function is unchanged, we're also dividing by that conjugate, so we're dividing by the same term. Which is square root of X2 + 4 X. Plus square root of X2 minus 3 X. Now for the numerator, we're going to apply the formula which Corresponds to the factorization of the difference of squares in reverse. Basically, this product corresponds to the square of the first term, which is X2 + 4 X minus the square of the second term. Which is x2 minus 3 X. Our denominator. is not affected, right? We're not trying to simplify anything here. We can just remove reCs because we're not multiplying by anything. And then we can just focus on the numerator, so X2 minus X is going to give us. No X squared terms in the numerator anymore. And we can just combine the like terms and the numerator, which is going to give us limit as X approaches infinity. Of 4 X minus -3 X, that's 7 X. And we are dividing by square root of X2 + 4 X. Plus square root of X2 minus 3 X. Now if we substitute infinity for acts we're going to get an indeterminate form of infinity divided by infinity. But because we have the irrational expression, what we're going to do is simply divide all of our terms by X. Because that's the highest exponent of X that we have among all of the independent variables in the resultant function. So we get a limit of 7. Because 7 X divided by X gives the 7. Divided by square root of, and if we're dividing by X, that's square root of X2 because X is bad stuff. So we're basically going to get square root of 1 + 4 divided by X. Under the radical, we're dividing everything by X squared. And now for the other radical, X2 divided by X gives us 1, and we're subtracting 3 x divided by X2, which is 3 divided by X. Now our fractional terms are going to approach 0. Because a non-zero number divided by infinity gives us a limit of 0. And overall, we get 7 divided by square root of 1 plus square root of 1. Which is seven halves. Well then, so we have got our answer which corresponds to the answer to the. 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 + 3x) − √(x² − 2x))

Key Concepts

Limits at Infinity

Conjugate Multiplication

Simplifying Expressions

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