Limits of quotients

Find the...

Question

Limits of quotients

Find the limits in Exercises 23–42.

limx→1 (x −1) / (√(x + 3) − 2)

Accepted Answer

Welcome back everyone to another video. Find the limit of the function below. Limit as x approaches 5 of x minus 5 divided by square root of X + 11 minus 4. So let's rewrite the limit. Limit as x approaches 5 of X minus 5 divided by square root of. X plus 11 minus 4. 1st of all, let's set on direct substitution, assuming that our function is continuous at X equals 5. We end up with 5 minus 5 in the numerator, and for the denominator, we're going to get square root of 5 + 11 minus 4. If we evaluate each part of this fraction, we're going to end up with 0 divided by 0, which is an indeterminate form. So one of the ways to solve for the limit as X approaches a number, not infinity, right, is to multiply both sides of our fraction by the conjugate of the part that contains square root, right? So we're going to rewrite our limit as limit as. X approaches 5. Now let's begin analyzing the denominator. It is square root of x + 11 minus 4, and what we're going to do is simply multiply it by the conjugate, which means that we're taking the same term with a positive sign instead of -4. We're going to take 4, so we're multiplying by a square root of X + E11 + 4, right? And because we want to make sure that our fraction is unchanged, we're also multiplying our numerator by the same conjugate so square root of X + 11 + 4. And from here we're just going to multiply. We're going to get limit as X approaches 5. Now, let's begin with the denominator. Here we have the factored form of the difference of squares, right? So we're going to square the first term, which gives us X plus 111. And then we're going to subtract the square of the second term, so we're subtracting for squad which is 16. For the numerator, we can just leave it as X minus 5 multiplied by square root of X + E11 + 4. And now let's calculate the result in the denominator X plus 11 minus 16 gives us x minus 5, and what we can notice is that now we can cancel out the common factor, right, X minus 5, which essentially allows us to rewrite the limit as limit as x approaches 5 of square root of. X + 11 + 4. And from here once again we can attempt the direct substitution. We're going to get square root of 5 + 11 + 4. Now square root of 16 is 4 and 4 + 4 gives us 8, which is our final answer to this problem. Thank you for watching.

Limits of quotients

Find the limits in Exercises 23–42.

limx→1 (x −1) / (√(x + 3) − 2)

Key Concepts

Limits

Quotient of Functions

Rationalization

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