If f(x) = x²+x+≤ft|x|/x, then what is ₓ → 0} f(x) equal to ?

Correct Answer is : lim⁡x→0ₓ → 0}x→0lim f(x) does not exist

Correct Answer is : lim⁡x→0ₓ → 0}x→0lim f(x) does not exist

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UPSC NDA 2 2022 Mathematics Solved Paper

Question : 86 of 120

Marks: +1, -0

If f(x) =

\frac{x^2+x+\left|x\right|}{x}

\lim\limits_{x \to 0}

f(x) equal to ?

Solution:

Concept:

The modulus function

\left|x\right|

is defined as:

\left|x\right| = x

when

x \ge 0

and

\left|x\right| = -x

when

x < 0

A limit

\lim_{x \to a} f(x)

exists only if the left‑hand limit equals the right‑hand limit.

Explanation:

Given

f(x) = \frac{x^{2} + x + \left|x\right|}{x}

For

x > 0

\left|x\right| = x

, so

f(x) = \frac{x^{2} + x + x}{x} = \frac{x^{2} + 2x}{x} = x + 2

Hence,

\lim_{x \to 0^{+}} f(x) = \lim_{x \to 0^{+}} (x + 2) = 0 + 2 = 2

For

x < 0

\left|x\right| = -x

, so

f(x) = \frac{x^{2} + x - x}{x} = \frac{x^{2}}{x} = x

Hence,

\lim_{x \to 0^{-}} f(x) = \lim_{x \to 0^{-}} x = 0

Since the right‑hand limit (

2

) is not equal to the left‑hand limit (

0

), the two‑sided limit does not exist.

Answer:

Option D:

\lim_{x \to 0} f(x)

does not exist.

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