Question 24 of 70intermediate🔧 ApplyShort Answer2 marks

Find lim⁡x→0f(x)\lim_{x \to 0} f(x), where f(x)={x∣x∣,x≠00,x=0f(x) = \begin{cases} \frac{x}{|x|}, & x \neq 0 \\ 0, & x = 0 \end{cases}

Correct Answer

The limit does not exist.

For finding the limit, we need to evaluate the left hand limit and right hand limit separately.

When x < 0 (approaching from left): |x| = -x, so f(x) = x/|x| = x/(-x) = -1. Therefore, lim(x→0⁻) f(x) = -1.

When x > 0 (approaching from right): |x| = x, so f(x) = x/|x| = x/x = 1. Therefore, lim(x→0⁺) f(x) = 1.

Since the right and left hand limits are different (-1 ≠\neq 1), the limit of f(x) as x tends to zero does not exist.

Exercise: EXERCISE 12.1 | Q: 26 | (Page Page 22)
For More Understanding

Explanation

This question tests the concept of one-sided limits. The context from Illustration 9 demonstrates that when left and right hand limits differ, the overall limit does not exist. The function x/|x| equals -1 for all negative x and 1 for all positive x, creating different one-sided limits at x = 0.

Solution Steps

  1. Step 1: Evaluate left hand limit by substituting x < 0, giving f(x) = x/(-x) = -1

  2. Step 2: Evaluate right hand limit by substituting x > 0, giving f(x) = x/x = 1

  3. Step 3: Compare both limits - since they differ, limit does not exist

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