NIMCET 2023 — Mathematics PYQ

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Question

NIMCET 2023 — Mathematics PYQ

NIMCET | Mathematics | 2023

Let f(x) = $\frac{( x ^{2} - 1 )}{( ∣ x ∣ - 1 )}$ . Then the value of $lim_{x \to - 1}$ f(x) is

Choose the correct answer:

Explanation

1. Analyzing the Function:

The given function is:

f(x) = \frac{x^2 - 1}{|x| - 1}

We need to find the limit as $x \to -1$ .

2. Behavior near $x = -1$ :

When $x$ approaches $-1$ , $x$ is a negative number. By the definition of absolute value:

|x| = -x \quad \text{for } x < 0

So, for the neighborhood of $-1$ , we can substitute $|x|$ with $-x$ .

3. Simplifying the Expression:

f(x) = \frac{x^2 - 1}{-x - 1}

Factorize the numerator using the identity $a^2 - b^2 = (a-b)(a+b)$ :

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

4. Canceling the common factor:

As $x \to -1$ , $x \neq -1$ , so we can cancel $(x + 1)$ :

f(x) = \frac{x - 1}{-1}

f(x) = -(x - 1) = 1 - x

5. Applying the Limit:

\lim_{x \to -1} f(x) = \lim_{x \to -1} (1 - x)

Substitute $x = -1$ :

\lim_{x \to -1} f(x) = 1 - (-1)

\lim_{x \to -1} f(x) = 1 + 1 = 2

Correct Option: 3

Choose the correct answer:

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