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NDA 2025
Exam
NDA
Year
2025
Subject
Mathematics
Chapter
Continuity and Differentiability

NDA 2025 · Mathematics · Continuity and Differentiability

Consider the following for the two (02) items that follow: Let f(x)...

Consider the following for the two (02) items that follow:
Let f(x)=f(x) =\begin{cases} x^3, & x^2 < 1 \\ x^2, & x^2 \geq 1 \end{cases}$.<br><br>Whatis.<br><br>What is\lim_{x \to 0} f'(x)$ equal to?

Options

  1. A22
  2. B11
  3. C00
  4. DLimit does not exist

Correct answer

C. 00

Step-by-step solution

Given f(x)=f(x) =\begin{cases} x^3, & x^2 < 1 \\ x^2, & x^2 \geq 1 \end{cases}$<br><br>For<br><br>Forx \to 0,therelevantdomainis, the relevant domain isx^2 < 1,whichmeans, which means-1 < x < 1.<br><br>Inthisinterval,thefunctionis.<br><br>In this interval, the function isf(x) = x^3.<br><br>Differentiatingwithrespectto.<br><br>Differentiating with respect tox,weget:<br><br>, we get:<br><br>f'(x) = 3x^2<br><br>Takingthelimitas<br><br>Taking the limit asx \to 0:<br><br>:<br><br>\lim_{x \to 0} f'(x) = \lim_{x \to 0} 3x^2 = 0<br><br>Answer:<br><br>Answer:0$
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