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 + 25) − √(x² − 1))

Accepted Answer

Below there today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the limit. The limit as X approaches infinity of parentheses, the square root of X2 + 36 minus the square root of X2 minus 4. Awesome. So it appears for this particular problem, we're ultimately trying to determine the limit that is provided to us. We're trying to determine the value of the limit. Awesome. So with that said, let's read off our multiple choice answers to see what our final answer might be. A is positive infinity, B is 0, C is negative infinity, and D is DNE, which means does not exist. Awesome. So with that said, first off, we need to recognize that the direct substitution of X equals positive infinity will lead to an indeterminate form, and because it leads to an indeterminate form, That will mean that We need to do a little bit of extra work in order to properly evaluate this limit, so let us quickly recognize the fact that The limit as X approaches infinity of the square root of X2 + 36 minus the square root of X2 minus 4. is equal to square brackets of the square root of infinity square + 36 minus the square root of infinity squared. -4. Will be equal to square brackets the square root of infinity minus the square root of infinity, which will be equal to square brackets of infinity minus infinity. Fantastic. So that's our indeterminate form. Awesome. So with that said, we now need to multiply and divide the expression by its conjugate and then simplify. So we need to go ahead and take the limit as X approaches infinity of the square root of X2 + 36 minus the square root of X2 minus 4. Multiplied by the square root of X2 + 36, plus the square root of X2 + minus 4, so it's gonna be plus the square root of X2 minus 4 divided by The square root of X2 + 36, plus the square root of X2 minus 4. Fantastic. So when we perform that multiplication, so when we multiply and divide the expression by its conjugate, and we simplify, I'm going to skip a few steps, but when you simplify down to its most simple form, you will find that the limit as X approaches infinity. Of 40. Divided by X, all divided by the square root of 1 plus. 36 divided by X to the power of 2. Fantastic. Plus, the square root of 1 minus 4 divided by X 2. Fantastic. So, once again, we take our initial limit, which is the limit as X approaches infinity of the square root of X2 + 36 minus the square root of X2 minus 4, and we are multiplying it and dividing the expression by its conjugate. And when we do that and we simplify it down to its most simple form, we should get that the limit as x approaches infinity of 40 divided by X, all divided by the square root of 1 + 36 divided by X2 plus the square root of 1 minus 4 divided by X2 as our answer. So now our next step is we need to evaluate the simplified limit expression as X approaches infinity. So we need to note that as X approaches infinity, the terms 40 divided by X, 36 divided by X2, and 4 divided by X to the 2 or X2 will approach 0. So we could go ahead and write that the limit as X approaches infinity of. 40 divided by X, divided by drum roll, the square root of 1 + 36 divided by X2, plus the square root of 1 minus. 4 divided by X 2 is going to be equal to square brackets of 0 divided by the square root of 1 + 0 plus the square root of 1 minus 0. Fantastic. And when we perform that calculation, we will find that our final final answer will have to be equal to 0. So therefore, our limit is 0, and that has to be our final answer. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be multiple choice answer letter B, which once again is 0. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Finding Limits of Differences When x → ±∞

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

lim x → ∞ (√(x² + 25) − √(x² − 1))

Key Concepts

Limits at Infinity

Conjugate Multiplication

Simplification of Radical Expressions

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