Test Index

UPSC NDA II 2023 Mathematics Solved Paper

Question Numbers: 57-58

Consider the following for the next items that follow:

Let f(x) = | x | and g(x) = [x] - 1, where [.] is the greatest integer function.

Let

h(x)=\frac{f(g(x))}{g(f(x))}

Question : 58 of 120

Marks: +1, -0

What is

\underset{x→0-}{lim}

h(x) equal to ?

Solution:

Concept:

The greatest integer function

[x]

gives the largest integer less than or equal to

x

Explanation:

We need to evaluate

\displaystyle\lim_{x\to0^-} h(x)

where

h(x)=\frac{|[x]-1|}{[|x|]-1}

x\to0^-

x

is a small negative number (for example,

x=-0.1

Then

[x] = -1

[x]-1 = -1-1 = -2

, so

|[x]-1| = |-2| = 2

Also

|x| = -x

(since

x<0

), so

[|x|] = [-x]

For

x\to0^-

-x

is a small positive number slightly above

0

, so

[-x] = 0

Thus

[|x|]-1 = 0-1 = -1

Therefore the limit becomes

\frac{2}{-1} = -2

Answer:

\displaystyle\lim_{x\to0^-} h(x) = -2

(Option A).

Go to Question:

Prev Question

Next Question