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UPSC NDA I 2016 Mathematics Solved Paper

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Question Numbers: 46-47

For the next two (2) items that follow:

Consider the function

f(x)=\frac{a^{[x]+x}-1}{[x]+x}

where [.] denotes the greatest integer function.

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Question : 46 of 120

Marks: +1, -0

What is

\lim\limits_{x\to 0^+}f(x)

equal to?

1
In a
$1 - a^{-1}$
Limit does not exist

Solution:

Concept:

Use the greatest integer property: for

x \to 0^+

,

[x] = 0

, and the standard limit

\lim\limits_{x \to 0} \frac{a^x - 1}{x} = \ln a

.

Explanation:

Given

f(x) = \frac{a^{[x] + x} - 1}{[x] + x}

.

We need

\lim\limits_{x \to 0^+} f(x)

.

For

x \to 0^+

,

x

is a very small positive number.

Thus

[x] = 0

because

0 \le x < 1

.

Substitute

[x] = 0

into

f(x)

:

f(x) = \frac{a^{0 + x} - 1}{0 + x} = \frac{a^x - 1}{x}

.

Now

\lim\limits_{x \to 0^+} f(x) = \lim\limits_{x \to 0^+} \frac{a^x - 1}{x}

.

This is the standard limit, which equals

\ln a

.

Answer:

\ln a

(Option B).

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