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UPSC NDA 2 2021 Mathematics Solved Paper

Question : 74 of 120

Marks: +1, -0

What is

\displaystyle\lim\limits_{n\to\infty} \frac{a^n+b^n}{a^n-b^n}

where a > b > 1, equal to?

Solution:

Concept:

The limit evaluates to 1 because the dominant term

a^n

in numerator and denominator cancels, and the ratio

(b/a)^n

tends to 0 when

a > b > 1

Explanation:

We are given

\displaystyle\lim_{n\to\infty} \frac{a^n+b^n}{a^n-b^n}

with

a > b > 1

Factor

a^n

from both numerator and denominator:

\displaystyle\lim_{n\to\infty} \frac{a^n\left[1+\left(\frac{b}{a}\right)^n\right]}{a^n\left[1-\left(\frac{b}{a}\right)^n\right]} = \lim_{n\to\infty} \frac{1+\left(\frac{b}{a}\right)^n}{1-\left(\frac{b}{a}\right)^n}

Since

0 < \frac{b}{a} < 1

, we have

\left(\frac{b}{a}\right)^n \to 0

n\to\infty

Thus the limit becomes

\frac{1+0}{1-0} = 1

Answer:

C. 1

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