NIMCET 2010 — Mathematics PYQ
NIMCET | Mathematics | 2010If f(x)={xsin(x1)0amp;for x=0amp;for x=0, then:
Choose the correct answer:
- A.
f is a continuous function
- B.
f′(0+) exists but f′(0−) does not exist
- C.
f′(0+)=f′(0)
f′(0+) and f′(0−) do not exist
Explanation
Solving:
1. For Continuity at x=0:
x→0limf(x)=x→0limxsin(x1)
Since −1≤sin(x1)≤1:
0×(finite value between −1 and 1)=0
∵limx→0f(x)=f(0)=0
∴f(x) is continuous.
2. For Differentiability at x=0:
f′(0)=h→0limhf(0+h)−f(0)
f′(0)=h→0limhhsin(h1)−0
f′(0)=h→0limsin(h1)
Limit does not exist (oscillates between −1 and 1).
Explanation
Solving:
1. For Continuity at x=0:
x→0limf(x)=x→0limxsin(x1)
Since −1≤sin(x1)≤1:
0×(finite value between −1 and 1)=0
∵limx→0f(x)=f(0)=0
∴f(x) is continuous.
2. For Differentiability at x=0:
f′(0)=h→0limhf(0+h)−f(0)
f′(0)=h→0limhhsin(h1)−0
f′(0)=h→0limsin(h1)
Limit does not exist (oscillates between −1 and 1).