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UPSC NDA I 2023 Mathematics Solved Paper

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Question : 100 of 120

Marks: +1, -0

What is

\underset{x→1}{lim}\frac{x^{9}-1}{x^{3}-1}

equal to ?

$-1$
$-3$
3
Limit does not exist

Solution:

Concept:

Use the algebraic identity

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

to factor both numerator and denominator.

Explanation:

Write the limit as

\underset{x \to 1}{\lim} \frac{x^9 - 1}{x^3 - 1}

.

The numerator is

(x^3)^3 - 1^3

, so it factors as

(x^3 - 1)(x^6 + x^3 + 1)

.

Cancel the common factor

(x^3 - 1)

from numerator and denominator.

The expression simplifies to

\underset{x \to 1}{\lim} (x^6 + x^3 + 1)

.

Substitute

x = 1

:

1^6 + 1^3 + 1 = 1 + 1 + 1 = 3

.

The limit exists and equals

3

.

Answer:

3

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