Limits of quotients

Find the...

Question

Limits of quotients

Find the limits in Exercises 23–42.

limu→1 (u⁴ − 1)/(u³ − 1)

Accepted Answer

Welcome back, everyone. Determine the limit of the function below. We want to identify the limit as X approaches 2 of X cubed minus 8 divided by X2 minus 4. Generally, we begin solving for limits by assuming that our function is continuous at the given point. In this case, we're going to perform direct substitution of X equals 2 to see if this is the case. We're going to get 2 cubed minus 8 divided by 22 minus 4. And as we understand, This leads to an indeterminate form which is 0 divided by 0. Therefore, what we want to do is perform some analysis on this function, and we can notice that our numerator and the denominator can be both factored out. So let's perform factorization to potentially eliminate a whole in the function and see if we can arrive at the final limit. So now first of all, let's notice that X cubed minus 8 is a difference of of cubes, X cubed minus 2 cubed. And now let's recall that if we have a difference of two cubes, such as a cubed minus B cubed, we can factor it out as a minus B multiplied by A2d plus AB. Plus be squared. We're going to use this formula in our case, A is X and B is 2. So this gives us X minus 2 multiplied by X2 plus AB, so that's 2 X plus B2, which is 2 squad and that's 4. And now for the denominator, we have a difference of squares. X2 minus 4 is X2 minus 22, so we're going to perform factorization as a minus B multiplied by A + B, using the A2 minus B2 formula. In this case, once again, A is X and B is 2, so we're going to get X minus 2 multiplied by X + 2. Now let's write those factored forms in our limit. In the numerator, we're going to get X minus 2 multiplied by X2 + 2 x + 4. In the denominator X minus 2 multiplied by x + 2. From here we can see that we can cancel out a common factor which is X minus 2, and we can once again attempt the direct substitution for the remaining part. Let's consider the numerator. We're going to get 2 squared plus 2 multiplied by 2 + 4, and in the denominator that's 2 + 2. In the numerator, we simply have 4 + 4 + 4, that's 12, in the denominator we have 4. And when we divide 12 by 4, we get 3 as a finite value which corresponds to our final limit because it's not an indeterminate form anymore. Thank you for watching.

Limits of quotients

Find the limits in Exercises 23–42.

limu→1 (u⁴ − 1)/(u³ − 1)

Key Concepts

Limits

Quotient of Functions

Factoring Polynomials

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