Calculate the following limits using the factorization formula xn−an=1x−a21xn−1+axn−2+a2xn−3+…+an−2x+an−12, where n is a positive integer a is a real number. limx→axn−anx−a, for any positive integer n

Resolved: "Calculate the following limits using the factorization formula..."

glamrockqueen7

2021-11-01

Calculate the following limits using the factorization formula

x^{n} - a^{n} = 1 x - a 21 x^{n - 1} + a x^{n - 2} + a^{2} x^{n - 3} + \dots + a^{n - 2} x + a^{n - 1} 2

, where n is a positive integer a is a real number.

lim_{x \to a} \frac{x^{n} - a^{n}}{x - a}

, for any positive integer n

jlo2niT

Skilled2021-11-02Added 96 answers

Step 1
To compute the limit

lim_{x \to a} \frac{x^{n} - a^{n}}{x - a}

.
We have

(x^{n} - a^{n}) = (x - a) (x^{n - 1} + a x^{n - 2} + \dots + a^{n - 2} x + a^{n - 1})

.
Step 2
Substitute in limit and compute as follows:

lim_{x \to a} \frac{x^{n} - a^{n}}{x - a} = lim_{x \to a} \frac{1}{x - a} ((x - a) (x^{n - 1} + a x^{n - 2} + \dots + a^{n - 2} x + a^{n - 1}))

= lim_{x \to a} (x^{n - 1} + a x^{n - 2} + \dots + a^{n - 2} x + a^{n - 1})

= a^{n - 1} + a^{n - 1} + \dots + a^{n - 1}

= n a^{n - 1}

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