Calculate the following limits using the factorization formula x^n−a^n=(x−a)(x^n−1+ax^n−2+a^2x^n−3+⋯+a^n−2x+a^n−1), where n is a positive integer and a is a real number.

lim x→1 x^6 − 1 / x − 1

Video transcript

Welcome back, everyone. Determine the limit using the factorization formula for polynomials. We want to evaluate the limit of X4 minus 16 divided by X minus 2 S X approaches 2, and we're given four answer choices A says 32, B 30, C 20. 8 and 26. So if we want to use the factorization formula for polynomials, we have to recall that whenever we have X to the power of m and we are subtracting a to the power of M, where A is a number, we can perform factorization as X minus a multiplied by x the power of N minus 1. Plus a multiplied by x to the power of n minus 2. And so on until we arrive at a to the power of n minus 2 X plus a the power of N minus 1. So essentially the final term will not have any X, right? So the exponent of A is increasing by 1 unit each time. The exponent of X is decreasing. So for this specific example, we have X minus 16 and 16 is 24th. So we have X, which is X, and A, which is 2. So let's use the formula. We end up with X minus 2 for the first factor, and now for the second factor, we can see that first of all, we take X to the power of N minus 1 and N is 4. So N minus 1 gives us 3, so we get X cubed. Plus A and now A is 2. X to the power of and minus 2, so 4 minus 2 gives us 2. We get 2 X2 plus. Now we get a. To the power of m minus 2, and that would be 4 minus 2, which gives us 2 and 2 squared is 4. And then we end up with X. To the power of m minus. 3 because we had n minus 2 previously and we have to decrease that exponent by 1 unit once again. So that be 4 X, which essentially corresponds to the penultimate term, right, which still has X and then finally we have 8 the power of n minus 1. So 2 to the power of 3 because 4 minus 1 gives us 3 and we get 8. So we have successfully factorized. They give an expression and now let's go back to the limit. X to the power of 4 minus 16 is going to be rewritten as X minus 2 multiplied by XQ plus 2 X2 + 4 X + 8. This is going to be divided by x minus 2, and this allows us to cancel out x minus 2 as a common factor, and then we can apply direct substitution since we only have the numerator. So we get 2 cubed plus 2 multiplied by 2 squared plus 4 multiplied by 2 + 8. Notice that we have 2 cubed plus 2 cubed plus 8 + 8, and since 2 cubed is 8. We simply have 8 + 8 + 8 + 8, which is 8 multiplied by 4, and that's 32. It is a finite number, so we have successfully evaluated the limit and the correct answer corresponds to the answer choice A. Thank you for watching.