What is ₓ arrow 0^{-}} h(x)+ ₓ arrow 0+} h(x) equal to ?

Correct Answer is : −32-3/2−23

Correct Answer is : −32-3/2−23

Test Index

UPSC NDA I 2024 Mathematics Paper

Question Numbers: 87-88

Consider the following for the items that follow :

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

Let

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

Question : 88 of 120

Marks: +1, -0

What is

\displaystyle \lim _{x \rightarrow 0^{-}} h(x)+\lim _{x \rightarrow 0+} h(x)

equal to ?

Solution:

Concept:

We compute the left-hand limit and right-hand limit of

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

x \to 0

, then add them.

Explanation:

Given

f(x) = |x| + 1

g(x) = [x] - 1

(where

[.]

is the greatest integer function), and

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

For

x \to 0^{-}

(left side):

|x| = -x

, so

f(x) = -x + 1

;

[x] = -1

, so

g(x) = -1 - 1 = -2

Thus

h(x) = \frac{-x+1}{-2} = \frac{x-1}{2}

x \to 0^{-}

h(x) \to \frac{0-1}{2} = -\frac{1}{2}

For

x \to 0^{+}

(right side):

|x| = x

, so

f(x) = x + 1

;

[x] = 0

, so

g(x) = 0 - 1 = -1

Thus

h(x) = \frac{x+1}{-1} = -(x+1)

x \to 0^{+}

h(x) \to -(0+1) = -1

Now add:

\displaystyle \lim_{x \to 0^{-}} h(x) + \lim_{x \to 0^{+}} h(x) = -\frac{1}{2} + (-1) = -\frac{3}{2}

Answer:

-\frac{3}{2}

(Option A).

Go to Question:

Prev Question

Next Question