What is {x→0}{lim} f(x) equal to ?

Correct Answer is : −23-2/3−32

Correct Answer is : −23-2/3−32

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UPSC NDA II 2023 Mathematics Solved Paper

Question Numbers: 63-64

Consider the following for the next items that follow:

Let

f(x)=\begin{cases} ax(x+1)+b, & x<1 \\ x-1, & 1\le x\le 2 \end{cases}

Question : 64 of 120

Marks: +1, -0

What is

\underset{x→0}{lim}

f(x) equal to ?

Solution:

Concept:

For

x \to 0

, use the branch of the piecewise function where

x<1

Explanation:

Given

f(x) = \begin{cases} ax(x+1)+b, & x<1 \\ x-1, & 1 \le x \le 2 \end{cases}

Since

x \to 0

lies in the interval

x < 1

, the relevant expression is

f(x) = ax(x+1) + b

Thus,

\underset{x \to 0}{\lim} f(x) = \underset{x \to 0}{\lim} [ax(x+1) + b]

Substitute

x = 0

directly:

a \cdot 0 \cdot (0+1) + b = 0 + b = b

From the earlier part of the problem (ensuring continuity at

x=1

), we have

b = -\frac{2}{3}

Therefore,

\underset{x \to 0}{\lim} f(x) = -\frac{2}{3}

Answer:

-\frac{2}{3}

(Option B).

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