AP EAMCET20224 Jul 2022Evening ShiftMathematicsContinuity and DifferentiabilityActual
If f x = x 2 ln cos x ln 1 + x 2 , x ≠ 0 0 , x = 0 , then f x is
Options
- Adiscontinuous at zero
- Bcontinuous but not differentiable at zero
- Cdifferentiable at zero
- Dnot continuous and not differentiable at zero
Correct answer
C. differentiable at zero
Step-by-step solution
f ' 0 + = lim h → 0 f 0 + h - f 0 h = lim h → 0 f h h = lim h → 0 ln cos h ln 1 + h 2 × h 2 h 2             . . . . 1 Multiplying Numerator & denominator by h = lim h → 0 h 2 ln 1 + h 2 × lim h → 0 ln cos h h 2 = lim h → 0 h 2 ln 1 + h 2 × lim h → 0 - tan h 2 h Using L'Hospital's Rule = 1 × - 1 2 = - 1 2 and f ' 0 - = lim h → 0 f 0 - h - f 0 - h = lim h → 0 f - h - h = - lim h → 0 - h ln cos