What is ₓ→ 0^{-}}f(x) equal to? [NDA I 2016]

Correct Answer is : 1−α−11-α^{-1}1−α−1

Correct Answer is : 1−α−11-α^{-1}1−α−1

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UPSC NDA I 2016 Mathematics Solved Paper

For the next two (2) items .that follow :

Consider the function

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

where [·] denotes the greatest integer function.

[NDA I 2016]

Question : 47 of 120

Marks: +1, -0

What is

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

Solution:

\lim\limits_{x\to 0^{-}}f(x)=\lim\limits_{h\to 0}\left[\frac{a^{[0-h]+(0-h)}-1}{[0-h]+(0-h)}\right]

=\lim\limits_{h\to 0}\frac{a^{[-h]-h}-1}{[-h]+(-h)}

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

=\frac{a^{-1-0}-1}{-1-0}=\frac{a^{-1}-1}{-1}

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

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