Evaluate lim x→1 3√x − 1 / x (Hint: x−1=(3√x)^3−1^3.)

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Accepted Answer

Welcome back, everyone. Determine the following limit. Limit as X approaches 27 of cubic root of X minus 3 divided by X minus 27. We're given for our answer choices AS 27, B 1 divided by 27, C 1 divided by 18, and D 18. So essentially, we are given a limit, which means that we have to begin solving this problem by Attempting direct substitution. We have cubic root of X minus. We divided by x minus 27 and we have to substitute 27 for every X so we get cubic root of 27 minus 3 divided by 27 minus 27. And this gives us an indeterminate form 0 divided by 0. Which means that we have to try and. We arrange the given function perhaps by factoring it out, and we can notice that the denominator can be rewritten as the difference of cubes. Let's understand that x minus 27. Can also be understood as x the power of 1/3 cubed. Which basically gives us X, right? Minus 3 cubed and this is a difference of 2 cubes. We can rewrite the difference of two cubes, a cubed minus B cubed in a factored form. A minus B multiplied by A2 + B + B2. So using this form, we end up with Cubic root of X, because X is the power of 1/3 is cubic root of X. Minus B, B is 3. Multiplied by a squared, so we basically have. Cubic root of X squared. Plus AB, which is 3. Cubic root of X. Let's B squad which is 3 squared and that's 9. And now we can rewrite the limit in a factored form. So we get Cubic root of X minus 3 in the numerator. In the denominator, we end up with cubic root of X. 3 multiplied by cubic root of X2. Plus 3 cubic root of X plus 9. And we can see that this is indeed useful because now we can cancel out one of the common factors. And finally, we can evaluate the limit, right, because we have one in the numerator. And for the denominator, we want to attempt direct substitution once again. So we get Cubic road Of 27 squared. Plus 3 cubic root of 27 + 9. Now cubic root of 27 is 3 and 3 squad. Gives us 9 so we end up with 1 divided by 9. Plus 3 multiplied by 3 gives us another 9, and then another 9 is added. So we end up with 1 divided by 27, which corresponds to the answer choice B. Thank you for watching.

Evaluate lim x→1 3√x − 1 / x (Hint: x−1=(3√x)^3−1^3.)

Key Concepts

Limits

Factoring and Rationalization

The Difference of Cubes Formula

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