84. Find lim(x→∞) (√(x² + 1) - √x).

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Accepted Answer

Welcome back, everyone. Evaluate the limit limit as X approaches infinity of square root of X2 + 2X minus X. A says 1, B2, C 0, and D infinity. For this problem, if we simply substitute X approaches infinity for every X, we're going to get square root of infinity minus infinity. That's basically an indeterminate form. We cannot really tell if that limit results in a finite value or not, right? So what we're going to do is recognize that we have a radical term minus X, and whenever this is the case, we're going to multiply and divide it by the conjugate. So we're going to get limit as X approaches infinity. We're going to rewrite the original function square root of X2 + 2X minus X. And we're going to multiply by its conjugates, so that would be the same expression, but with a positive sign, right? So we're going to take square root of X2 + 2 X plus X instead of minus X. And we're going to divide by the conjugate to ensure that the original function is unchanged. So we are dividing by square root of X2 + 2 X plus X. For the numerator, let's recognize that this is a factored form of the difference of squares. So multiplying, we're going to get square of the radical term that's simply X2 + 2X minus the square of the. The term which is X. So we're subtracting X2, and our denominator remains the same, right? It is going to be square root of X2 + 2 X plus X. In the numerator, we can cancel out x2d. And what we can do from here is again, check what's happening if X approaches infinity. In the numerator, we're going to get infinity in the denominator. Square root of infinity plus infinity again this is infinity. So we can perform factorization. Let's factor out X. In the numerator, we already have 2 X, so it is factored. And in the denominator, we're going to factor out X, which leaves us with square root of. X 2 divided by X2 gives us 1 and 2 X divided by X2 gives us 2 divided by X. Plus x divided by X gives us a 1. Now we can cancel out X in the numerator and in the denominator. And understand that as x approaches infinity, the fractional term approaches 02 divided by infinity gives us 0 because we have a constant divided by an infinitely large number. So now we can perform the calculation in the numerator we have 2. In the denominator, we have square root of 1 + 0, which is square root of 1 + 1. And that's 2 divided by 2. Which is equal to one, meaning the correct answer for this problem corresponds to the answer choice a. Thank you for watching.

84. Find lim(x→∞) (√(x² + 1) - √x).

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