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NDA 1 2024 Maths, question 88

Topic: Differential calculus

Passage · Q87 to Q88

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

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

Q88

What is lim⁡x→0−h(x)+lim⁡x→0+h(x)\underset{x\to 0-}{\lim }h(x)+\underset{x\to 0+}{\lim }h(x) equal to ?

  1. (a) −32-\frac{3}{2}Correct answer
  2. (b) −12-\frac{1}{2}
  3. (c) 12\frac{1}{2}
  4. (d) 32\frac{3}{2}

Solution.

Answer: (a) −32-\frac{3}{2}

  1. As x approaches 0 from below, floor(x) = −1, so g(x) = −2 and h(x) tends to −1/2.
  2. From above, floor(x) = 0, so g(x) = −1 and h(x) tends to −1.
  3. Their sum is −3/2.