Limits of quotients

Find the...

Question

Limits of quotients

Find the limits in Exercises 23–42.

limx→−1 (√(x² + 8) − 3) / (x + 1)

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the limit as X approaches 2 of the square root of X2 + 5 minus 3 divided by X minus 2. A says it's 2/3, B 4/3, C 3 houses, and D 3 quarters. Now, let's first try to determine the limit by direct substitution. We might have a problem when we do that, OK? Because if we directly substitute. X equals 2 into our limit, OK, then notice that our denominator will be 2 minus 2, which is 0, and we can't divide by 0. So by direct substitution of X minus 2, OK, then a result, it leads to. An indeterminate form, OK? And since it's in an indeterminate form that way, that means we have to try and find another way to rewrite or limit so that it doesn't lead to an indeterminate form. So how can we do that? Well, we could multiply the numerator and denominator by the conjugate of the numerator, and then we can try to simplify and then substitute X equals 2. So let's do that. So here, OK, I could also make a note of that. So by multiplying. I'm multiplying. By the conjugate. Or I could have just written multiplying by the conjugate. It's the same thing. Then We're going to have the limit. Let me make some space here. We're gonna take the limit as X approaches 2 of the square root of X2 + 5 minus 3 divided by X minus 2. And we're going to multiply it by the square root of X2 + 5 + 3. Divided by the square root of X2 + 5 + 3. So we're multiplying the numerator and denominator by the conjugate. Now, when we do that, OK, in our numerator. You're gonna have the limit as X approaches 2, OK, of X2 + 5, OK, minus 9. So that's what we're we'll have in our numerator, OK? And on the other hand, in our denominator, we're going to have X minus 2 multiplied by the square root of X2 + 5 + 3. Now if we take this a bit further here, OK, then we're going to have. Limit as X approaches 2 of X2 minus 4. Divided by our denominator here, which is X minus 2 multiplied by the square root of X2 + 5 + 3. And now if you think about it, we could rewrite. Let me, let me go down. We could actually rewrite our numerator, OK. As the difference of two squares, X minus 2 multiplied by X + 2, OK, divided by X minus 2 multiplied by the square root of X2 + 5 + 3. Now that's pretty useful because we can cancel out a factor of X minus 2. And then that's gonna leave us with the limit, OK, as X approaches 2 of X + 2 divided by the square root of X2 + 5 + 3. And now, in this case, we can try to evaluate the simplified limit expression as X approaches 2. So now, that means by substituting 2, this is gonna become 2 + 2. Divided by the square root of 22 + 5 + 3. 2 + 2 is 4, OK. 2 squared is 4 and 4 + 5 equals 9, so that's route 9 + 3. Route 9 E equals 3, so this is going to be 4 divided by 3 + 3, which is 6, and a 4/6 simplified is 2/3. Thus, the limit as X approaches 2 of the square of X2 + 5 minus 3 divided by X minus 2 equals 2/3. If we look back at our answer choices, that means A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Limits of quotients

Find the limits in Exercises 23–42.

limx→−1 (√(x² + 8) − 3) / (x + 1)

Key Concepts

Limits

Rational Functions

L'Hôpital's Rule

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